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**************************************************************************************************Mȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩȩ̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧̧9.V.0zPPRRRRRRRR<<<</(<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<KKKKKKKKKxD0,G,D0,G,D0,G,D0,G,D0,G,D0,G,Math::Pari::Not a matrix where matrix expectedMath::Pari::PARIcolMath::Pari::interface5Math::Pari::interface25Math::Pari::interface45Math::Pari::have_highlevelin, ...gne_gne_exptoo many iterations in roots2() ( laguer() ):
PolGGDGbnrisconductor(a1,{a2},{a3}): returns 1 if the modulus is the conductor of the subfield of the ray class field given by a1,a2,a3 (see bnrdisc), and 0 otherwise. Slightly faster than bnrconductor if this is the only desired resultcore(n,{flag=0}): unique (positive of negative) squarefree integer d dividing n such that n/d is a square. If (optional) flag is non-null, output the two-component row vector [d,f], where d is the unique squarefree integer dividing n such that n/d=f^2 is a squarecotan(x): cotangent of xderivellak(e,n): computes the n-th Fourier coefficient of the L-function of the elliptic curve eellidentifyforsubgroupkroneckermatcompanionmateigen(x): eigenvectors of the matrix x given as columns of a matrixmattransposeminpolynfdisc(x,{flag=0},{p}): discriminant of the number field defined by the polynomial x using round 4. Optional args flag and p are as in nfbasisnfeltpowmodprnfnewprecpolcyclo(n,{v=x}): n-th cyclotomic polynomial (in variable v)lGDnpolinterpolate(xa,{ya},{x},{&e}): polynomial interpolation at x according to data vectors xa, ya (ie return P such that P(xa[i]) = ya[i] for all i). If ya is omitter, return P such that P(i) = xa[i]. If present, e will contain an error estimate on the returned valueprimesprintpquadunit(x): fundamental unit of the quadratic field of discriminant x where x must be positivernfcharpoly(nf,T,alpha,{var=x}): characteristic polynomial of alpha over nf, where alpha belongs to the algebra defined by T over nf. Returns a polynomial in variable var (x by default)rnfisfree(bnf,order): given an order as output by rnfpseudobasis or rnfsteinitz, outputs true (1) or false (0) according to whether the order is free or notsqrt(x): square root of xsqrtnwrite1(filename,a): write the string expression a (same output as print1) to filenameznlog(x,g): g as output by znprimroot (modulo a prime). Return smallest non-negative n such that g^n = xgasininsufficient precision for generators, not givencompleting bnf (building cycgen)
*** Ideal no %ld: [%Z, %Z, %Z, %Z]
SubCyclo: conductor:%ld
wrong type in galoissubcycloalgdep(x,n)=algebraic relations up to degree n of xbilhell(e,z1,z2)=canonical bilinear form for the points z1,z2 on the elliptic curve e. Either z1 or z2 can also be a vector/matrix of pointsGD0,G,D0,G,D3,L,factpol2(x,l)=factorization over Z of the polynomial x up to degree l (complete if l=0) using root findingidealaddmultone(nf,list)=when the sum of the ideals in the number field K defined by nf and given in the vector list is equal to Z_K, gives a vector of elements of the corresponding ideals who sum to 1ideallistarch(nf,list,arch)=vector of vectors of all zidealstarinits of all modules in list with archimedean arch added, without generatorsindsort(x)=indirect sorting of the vector xisisom(x,y)=tests whether the number field defined by the polynomial x is isomorphic to the one defined by y; 0 if not, otherwise all the isomorphismsispspispsp(x)=true(1) if x is a strong pseudoprime, false(0) if notisqrtlll(x)=lll reduction of the vectors forming the matrix x (gives the unimodular transformation matrix)lllgen(x)=lll reduction of the vectors forming the matrix x with polynomial coefficients (gives the unimodular transformation matrix)mod(x,y)=creates the integer x modulo y on the PARI stackmodreverse(x)=reverse polymod of the polymod x, if it existsnumer(x)=numerator of xpolintpolredabsnoredregularhorealrhorealnodsigma(x)=sum of the divisors of xsignat(x)=signature of the symmetric matrix xsinh(x)=hyperbolic sine of xsumalt2(X=a,expr)=Cohen-Villegas-Zagier's acceleration of alternating series expr, X starting at asuminf(X=a,expr)=infinite sum (X goes from a to infinity) of real or complex expression exprtruncweipellzell(e,z)=In the complex case, lattice point corresponding to the point z on the elliptic curve ezidealstarinit(nf,I)=6-component vector [I,v,fa,f2,U,V] where v is as in zidealstar without the generators, fa is the prime ideal factorisation of I and f2, U and V are technical but essential to work in (Z_K/I)^*[rnfkummer] bnfinit(Kz)Step 4
Step 18
\vskip\smallskipamount$\displaystyle{\tt\%#1} = #2$expected character: ']'Rg_to_FlFpXQX_gcdFpM_invimagemodulargcd: trial division failedQXQ_inv: mod %ld (bound 2^%ld)gdeflateLLL_cmbf: (a,b) =%4ld,%4ld; r =%3ld -->%3ld, time = %ld
subresallgauss. i=%lddetint. k=%ldUnrecognized action in a templateorderstuunused characters: %sbreak not allowed here (reading function args)skipidentifier (unknown code)negative valuation in laplacegmul2nN0 in QuickPol: %ld 
polrelnumN0 = %ld
RecCoeff3: no solution found!
L(6) = PSL(2,5) = A_5(6)incorrect galois automorphism in galoisapplygp_readvec_stream: reaching %ld entries
unknown code in readobjmatrix(0,%ld)read failed%s written by an incompatible version of GPweipellnumnon polynomial or series type substituted in a seriesgfloor2ninvalid vector in qfevaltoo large negative arg %ld in zetagpolylogfor this traceincorrect modpr format  fastnu: G is computed
plotinitvLGGD0,L,plotlinetype(w,type): change the type of following lines in rectwindow w. type -2 corresponds to frames, -1 to axes, larger values may correspond to something else. w=-1 changes highlevel plottingplotscale(w,x1,x2,y1,y2): scale the coordinates in rectwindow w so that x goes from x1 to x2 and y from y1 to y2 (y2<y1 is allowed)M* = %Z
(lower bound for regulator) M = %Z
invalid flagimpossible assignment I-->Sincorrect dimensions in initrect%ld %ld p
%ld %ld m %ld %ld l
IFAC: untested integer declared primeMPQS: sieving interval = [%ld, %ld]
MPQS:   (%ld of these from partial relations)
 
\\ MATRIX READ BY MPQS
FREL=MPQS: splitting N after %ld kernel vector%s

[Entering compute_data()

incorrect denominator in initgaloisborne: %ZA4GaloisConj: %ld hop sur %ld iterations
incorrect variable in caradjmatrixqz when the first 2 dets are zerohnfall[2], li = %ldTerminal does not define justify_textCan't find Gnuplot drawing engine DLL,
	set GNUPLOT_DRAW_DLL environment variable to the name of the DLL,
	or install Perl module Term::Gnuplot, e.g., by running
		perl -MCPAN -e "install Term::Gnuplot"
	With Term::Gnuplot, if you don't have root access, consult
		perldoc -q "my own"
	alternatively, you can use an uninstalled version of Term::Gnuplot
	by running GP/PARI as
		env PERL5OPT=-Mblib=/directory/of/build/of/Term-Gnuplot gp
dependent vectors in lllfpin precision %ld
fincke_pohsttime [max,t12,loop,reds,fin] = [%ld, %ld, %ld, %ld, %ld]
invalid coefficients in rootsnon integral index in prodinf1non integral index in sumposNon trivial conditional class group.
  *** May miss solutions of the norm equation  errdelta = %Z
ratlift: bmax must be > 0, found
	bmax=%Z
Got the type 0x%x instead of CV=0x%x or GV=0x%x in %s, %i2.030518Math::Pari::allocatememUnsupported code '%.1s' in signature '%s' of a PARI functionarg1, arg2, arg3=0gcmp1_flexible_intn = 0listsortreducible modulus in factornf  *** %s: %sbernvecbittest(x,n): gives bit number n (coefficient of 2^n) of the integer x. Negative numbers behave as if modulo big power of 2bnfdecodemodulebnfsignunit(bnf): matrix of signs of the real embeddings of the system of fundamental units found by bnfinitbnrrootnumbercomponent(x,s): the s'th component of the internal representation of x. For vectors or matrices, it is simpler to use x[]. For list objects such as nf, bnf, bnr or ell, it is much easier to use member functions starting with "."GD0,L,dirmul(x,y): multiplication of the Dirichlet series x by the Dirichlet series yellapelleisnum(om,k,{flag=0}): om=[om1,om2] being a 2-component vector giving a basis of a lattice L and k an even positive integer, computes the numerical value of the Eisenstein series of weight k. When flag is non-zero and k=4 or 6, this gives g2 or g3 with the correct normalizationelletaellgenerators(E): if E is an elliptic curve as output by ellinit(), return the generators of the Mordell-Weil group associated to the curve. This function depends on the curve being referenced in the elldata database.elllseriesGPidealpowideleprincipalincgam(s,x,{y}): incomplete gamma function. y is optional and is the precomputed value of gamma(s)isprime(x,{flag=0}): true(1) if x is a (proven) prime number, false(0) if not. If flag is 0 or omitted, use a combination of algorithms. If flag is 1, the primality is certified by the Pocklington-Lehmer Test. If flag is 2, the primality is certified using the APRCL test.iGGlex(x,y): compare x and y lexicographically (1 if x>y, 0 if x=y, -1 if x<y)matkermatrixqz(x,p): if p>=0, transforms the rational or integral mxn (m>=n) matrix x into an integral matrix with gcd of maximal determinants equal to 1 if p is equal to 0, not divisible by p otherwise. If p=-1, finds a basis of the intersection with Z^n of the lattice spanned by the columns of x. If p=-2, finds a basis of the intersection with Z^n of the Q-vector space spanned by the columns of xmaxmodreverse(x): reverse polymod of the polymod x, if it existsnfeltdiveuc(nf,a,b): gives algebraic integer q such that a-bq is smallnfeltmulnfeltvalnormnorml2(x): square of the L2-norm of the vector xnumtoperm(n,k): permutation number k (mod n!) of n letters (n C-integer)padicprecpolcompositumpoldiscreduced(f): vector of elementary divisors of Z[a]/f'(a)Z[a], where a is a root of the polynomial fpolrootspadic(x,p,r): p-adic roots of the polynomial x to precision rpolsylvestermatrix(x,y): forms the sylvester matrix associated to the two polynomials x and y. Warning: the polynomial coefficients are in columns, not in rowsprecprime(x): largest pseudoprime <= x, 0 if x<=1primepi(x): the prime counting function pi(x) = #{p <= x, p prime}.qfbredqfrep(x,B,{flag=0}): vector of (half) the number of vectors of norms from 1 to B for the integral and definite quadratic form x. Binary digits of flag mean 1: count vectors of even norm from 1 to 2B, 2: return a t_VECSMALL instead of a t_VECreturn({x=0}): return from current subroutine with result xsumalt(X=a,expr,{flag=0}): Cohen-Villegas-Zagier's acceleration of alternating series expr, X starting at a. flag is optional, and can be 0: default, or 1: uses a slightly different method using Zagier's polynomialstaylorvectorsmallcaching Bernoulli numbers 2*%ld to 2*%ld, prec = %ld
non-positive integer argument in cxgammaGamma not defined for non-integral p-adic numberincorrect big number fieldregulator is zero.
KC2=%ld, KC=%ld, KCZ=%ld, KCZ2=%ld
Subcyclo: val=%ld
subcyclo_cyclicapell2(e,p)=computes a_p for the elliptic curve e using Jacobi symbolsassmatGD0.1,G,D0.1,G,D5,G,conductorofchar(bnr,chi)=conductor of the character chi on the ray class group bnrideallistunitarchgenimageisinclfastkerlaplacemathell(e,x)=gives the height matrix for vector of points x on elliptic curve e using theta functionsGGVVImatrixqz2nfval(nf,a,pr)=valuation of element a at the prime prreduceddiscrootmod(x,p)=roots mod p of the polynomial xrounderror(x)=maximum error found in rounding xseries(x,v)=convert x (usually a vector) into a power series with variable v, starting with the constant coefficientsetsearch(x,y)=looks if y belongs to the set x. Returns 0 if it is not, otherwise returns the index j such that y==x[j]simplify(x)=simplify the object x as much as possiblesmith(x)=Smith normal form (i.e. elementary divisors) of the matrix x, expressed as a vectorsubellthetanullk(q,k)=k'th derivative at z=0 of theta(q,z)[rnfkummer] Selmer group1/0 exponent in FpXQ_sqrtnFpM_kerpolint_triv2 (i = %ld)bound for resultant: 2^%ld
partial factorization is not meaningful hereQ_contentnot the same variables in sylvestermatrixreduceddiscsmithginvmodincorrect mask in vecextractSolving the triangular system
ZM_invnot a polynomial of degree 2 in quadhilbertpowsubFBquadrnfidealhermitecheckrnfplease apply bnfinit firstmissing units in %sgalois of degree higher than 11nsiso0t_FRAC*** hashcode = %lu
\frac{%s is set (%s), but is not a directorymat(%ld,%ld) = I/O: closing file %s (code %d) 
skipping directory %sdegree overflow in pow_monomevaluation overflow in sqrtnnot an integral curve in elllocalredpoint not on elliptic curveentering zlog, to find factor %Zincorrect polynomial in rnf function  entering Nilordrelative basis computed
plotpointsplotpointsizenot a vector of ideals in idealaddmultooneinconsistent data...
ECM: time = %6ld ms, B1 phase done, SQUFOF: blacklisting a = %ld on second cycle
	But it wasn't a pure power.
ECM: stack tight, using heap space
IFAC: trying Shanks' SQUFOF, will fail silently if input
      is too large for it.
IFAC: incorporating set of %ld factor(s)
LPRELMPQS: relations file truncated?!
 0MPQS: full relations file %s than expectedTR_POL, i = %ld/%ldSecond arg. must be integer in galoisconj4GaloisAnalysis:Nbtest=%ld,p=%ld,o=%ld,n_o=%d,best p=%ld,ord=%ld,k=%ld
hnf[1]. i=%ldsqred2nb lines > nb columns in hnfmodPartitions of %ld (%ld)
read_objectIncomplete curve name in ellsearchNo such elliptic curveqfi_unitEntering LLL
bitsize bound: %ld
bitsize Norm: %ld
isrealapprrefine_Finvalid polynomial in thue (need deg>2)C (bitsize) : %d
x <= %Z
Math::Pari::listPariArray index %i out of rangearg1, arg2, arg3, arg4gand_ggt_gsubLV=GGEpD0,L,D0,L, %2d: %s
ggvalnegationgexpolistput   %3ld fact. of degree %3ld
non-positive precision in factorpadicuse pari_warn for warningsgcdMod(x,y): creates 'x modulo y'.Pi=Pi(): the constant pi, with current precisionbesseljbesseln(nu,x): N-bessel function of index nu and argument xbitand(x,y): bitwise "and" of two integers x and y. Negative numbers behave as if modulo big power of 2bnfnarrow(bnf): given a big number field as output by bnfinit, gives as a 3-component vector the structure of the narrow class groupbnfreg(P,{tech=[]}): compute the regulator of the number field defined by the polynomial P. If P is a non-zero integer, it is interpreted as a quadratic discriminant. See manual for details about techbnrconductorofcharcoshcosh(x): hyperbolic cosine of xdilog(x): dilogarithm of xelladd(e,z1,z2): sum of the points z1 and z2 on elliptic curve eforstepforstep(X=a,b,s,seq): the sequence is evaluated, X going from a to b in steps of s (can be a vector of steps)galoisexportgetrand(): current value of random number seedidealaddtooneideallist(nf,bound,{flag=4}): vector of vectors L of all idealstar of all ideals of norm<=bound. If (optional) flag is present, its binary digits are toggles meaning 1: give generators; 2: add units; 4: give only the ideals and not the bid.intformal(x,{y}): formal integration of x with respect to the main variable of y, or to the main variable of x if y is omittedintnuminitgen(t,a,b,ph,{m=0},{flag=0}): initialize tables for  integrations from a to b using abcissas ph(t) and weights ph'(t). Note that  there is no equal sign after the variable name t since t always goes from  -infty to +infty, but it is ph(t) which goes from a to b, and this is not  checked. If flag = 1 or 2, multiply the reserved table length by 4^flag, to  avoid corresponding error.isfundamentalispower(x,{k},{&n}): true (1) if x is a k-th power, false (0) if not. If n is given and a k-th root was computed in the process, put that in n. If k is omitted, return the maximal k >= 2 such that x = n^k is a perfect power, or 0 if no such k exist.lengthlexmathessnfsnfpermtonum(vect): ordinal (between 1 and n!) of permutation vectpolrecipqfminim(x,{bound},{maxnum},{flag=0}): number of vectors of square norm <= bound, maximum norm and list of vectors for the integral and definite quadratic form x; minimal non-zero vectors if bound=0. flag is optional, and can be 0: default; 1: returns the first minimal vector found (ignore maxnum); 2: as 0 but uses a more robust, slower implementation, valid for non integral quadratic formsquadgenrnfidealreltoabs(rnf,x): transforms the ideal x from relative to absolute representationGGD2,L,setminusuntilweber(x,{flag=0}): One of Weber's f function of x. flag is optional, and can be 0: default, function f(x)=exp(-i*Pi/24)*eta((x+1)/2)/eta(x) such that (j=(f^24-16)^3/f^24), 1: function f1(x)=eta(x/2)/eta(x) such that (j=(f1^24+16)^3/f2^24), 2: function f2(x)=sqrt(2)*eta(2*x)/eta(x) such that (j=(f2^24+16)^3/f2^24)vsDGznstar(n): 3-component vector v, giving the structure of (Z/nZ)^*. v[1] is the order (i.e. eulerphi(n)), v[2] is a vector of cyclic components, and v[3] is a vector giving the corresponding generatorspsi of power seriesfactor base.
(more relations needed: %ld)
Subcyclo: borne=%Z
algdep2asin(x)=inverse sine of xbuchrealchptell(x,y)=change data on point or vector of points x on an elliptic curve according to y=[u,r,s,t]comprealraw(x,y)=Gaussian composition without reduction of the binary quadratic forms x and y of positive discriminantdiscrayabs(bnr,subgroup)=absolute [N,R1,discf] of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroupdiscrayabslistarchall(bnf,bound)=gives list of discrayabscond of all modules up to norm bound with all possible archimedean places arch in reverse lexicographic order, in a longvector formatfactpol(x,l,hint)=factorization over Z of the polynomial x up to degree l (complete if l=0) using Hensel lift, knowing that the degree of each factor is a multiple of hintgaloisconj(nf)=list of conjugates of a root of the polynomial x=nf[1] in the same number field, using p-adics, LLL on integral basis (not always complete)galoisconj1gotohell2idealdivexact(nf,x,y)=quotient x/y of two ideals x and y in HNF in the number field nf when the quotient is known to be an integral idealideallistunitarchimagecompl(x)=vector of column indices not corresponding to the indices given by the function imageincgam4lllgramint(x)=lll reduction of the lattice whose gram matrix is the integral matrix x (gives the unimodular transformation matrix)localred(e,p)=e being an ellliptic curve, returns [f,kod,[u,r,s,t],c], where f is the conductor's exponent, kod is the kodaira type for e at p, [u,r,s,t] is the change of variable needed to make e minimal at p, and c is the local Tamagawa number c_pGGGGplseriesell(e,s,N,A)=L-series at s of the elliptic curve e, where |N| is the conductor, sign(N) the sign of the functional equation, and A a cut-off point close to 1modulargcdnewtonpoly(x,p)=Newton polygon of polynomial x with respect to the prime pnfdiveucpoly(x,v)=convert x (usually a vector or a power series) into a polynomial with variable v, starting with the leading coefficientpolyrevprecrounderrorsetrand(n)=reset the seed of the random number generator to nsignunit(bnf)=matrix of signs of the real embeddings of the system of fundamental units found by buchinitsmallinitellsmithclean(z)=if z=[u,v,d] as output by smith2, removes from u,v,d the rows and columns corresponding to entries equal to 1 in dsmithpol(x)=Smith normal form (i.e. elementary divisors) of the matrix x with polynomial coefficients, expressed as a vectorsylvestermatrixthreetotwo2vecsort(x,k)=sorts the vector of vector (or matrix) x according to the value of its k-th componentzeta(s)=Riemann zeta function at sStep 5
Computing Newton sums: boldfgpolsym_genmatratliftnfgcdfor this block of tracesLLL_cmbf: chk_factors failedintegral from infty to infty or from -infty to -inftymattodiagonalFqM_kererror in parse_option_stringcan't kill thatonly functions can be aliased.%s = break not allowed after #quadhilbert needs a fundamental discriminantnegative degree in legendrenot a set in setintersect/C(6) = 6 = 3[x]2S6F_42(7) = 7:6coefget_red_G: starting LLL, prec = %ld (%ld + %ld)
)
[]
\pmatrix{
 %ld unnamed objects read. Returning then in a vectormod(\PARIbreak I/O: opening file %s (mode %s)
%s not written for a %ld bit architecturegsqrtnp-adic argument out of range in gsinFp_shanks, k = %ldIdeallistgevalpadic polylogarithmremaining modular factor(s): %ld
  f = %Z,
  a = %Z
, parameters: %Z^%ld
  f = %Znfhilbert not soluble at finite place: %Z
couldn't open dynamic symbol table of processplotkillplotrecthplotrecthrawpsplothraw[1]: nfhermitemod, i = %ldincorrect vector length in idealrednot an Abelian extension in rnfnormgroup?length (lg) overflowLucasModMPQS: first sorting at %ld%%, then every %3.1f%% / %3.1f%%

MPQS: giving up.
MPQS: Gauss done: kernel has rank %ld, taking gcds...
MPQS: no solutions found from linear system solver5th powerconjugate %ld: %Z
galoisconj4matrixqz_auxallhnfmod        all integer roots are double roots
indefinite invariant polynomial in gpoly() %ld^%ld
$$$$$ Tschirnhaus transformation of degree %ld: $$$$$
 )
panic: gnuplotTerminal does not define pointsizenegative discriminant in %sModified P: %Z
Initialization time = %ld
smallvectors looking for norm < %Z
FlxqX_safegcdispower for non-rational argumentsnot a tnf in thuec1 = %Z
c2 = %Z
Indice <= %Z
  c8  = %Z
floorr (precision loss in truncation)PARI: %.*s%*s%sMath::Pari::interface_flexible_voidMath::Pari::interface_flexible_longMath::Pari::interface30gcmpglt_gdiv_gle`%s' is not a Pari function name%ldV=GGIV=GGID%sour data takes %ld words out of %ld words on stack
p = 1 in Z_lvalremgcvtopno more room in this listmallocing NULL object in newblocleaving recover()
  ***   %s %s, please report %s %s %s %s.  [hint] you can increase GP stack with allocatemem()
0x%p:	0x%lx	%lu
besselh1(nu,x): H^1-bessel function of index nu and argument xbnfisintnormbnfisnorm(bnf,x,{flag=1}): Tries to tell whether x (in Q) is the norm of some fractional y (in bnf). Returns a vector [a,b] where x=Norm(a)*b. Looks for a solution which is a S-unit, with S a certain list of primes (in bnf) containing (among others) all primes dividing x. If bnf is known to be Galois, set flag=0 (in this case, x is a norm iff b=1). If flag is non zero the program adds to S all the primes : dividing flag if flag<0, or less than flag if flag>0. The answer is guaranteed (i.e x norm iff b=1) under GRH, if S contains all primes less than 12.log(disc(Bnf))^2, where Bnf is the Galois closure of bnfbnfsignunitbnrinitdireulerGGDnelltaniyamagammaideallog(nf,x,bid): if bid is a big ideal, as given by idealstar(nf,I,1) or idealstar(nf,I,2), gives the vector of exponents on the generators bid[2][3] (even if these generators have not been computed)idealminintcircintformallGDGD&lindep(x,{flag=0}): Z-linear dependencies between components of x. flag is optional, and can be 0: default, PSLQ; -1: using Hastad et al; -2: returns a non-trivial kernel vector (not integral in general); positive, and in that case should be between 0.5 and 1.0 times the accuracy in decimal digits of x, using a standard LLLmatdiagonal(x): creates the diagonal matrix whose diagonal entries are the entries of the vector xnfeltdiv(nf,a,b): element a/b in nfnfhnfmodnfsubfields(nf,{d=0}): find all subfields of degree d of number field nf (all subfields if d is null or omitted). Result is a vector of subfields, each being given by [g,h], where g is an absolute equation and h expresses one of the roots of g in terms of the root x of the polynomial defining nfpoldisc(x,{v}): discriminant of the polynomial x, with respect to main variable if v is omitted, with respect to v otherwisepolzagier(n,m): Zagier's polynomials of index n,mprecision(x,{n}): change the precision of x to be n (n C-integer). If n is omitted, output real precision of object xprodeuler(X=a,b,expr): Euler product (X runs over the primes between a and b) of real or complex expressionrandomrnfhnfbasisrnfisfreernfnormgrouproundlLsignsizedigit(x): maximum number of decimal digits minus one of (the coefficients of) xsubgrouplistsumnumalt(X=a,sig,s,{tab},{flag=0}): numerical summation of (-1)^X s from X = ceiling(a) to +infinity. Note that the (-1)^X must not be included. sig is either a scalar or a two-component vector coded as in intnum, and the  scalar part is larger than all the real parts of the poles of s. Uses intnum, hence tab is as in intnum. If flag is nonzero, assumes that the function to  be summed satisfies conj(f(z))=f(conj(z)), and then up to twice faster.sumpos(X=a,expr,{flag=0}): sum of positive series expr, the formal variable X starting at a. flag is optional, and can be 0: default, or 1: uses a slightly different method using Zagier's polynomialsvss*znorderlim, nn: [%ld, %ld], la = %lf
    Norm(P) > Zimmert bound
    #%ld in factor base
small norm relations
#### Computing check
abs(x)=absolute value (or modulus) of xbilhellconcat(x,y)=concatenation of x and ycoredisc2det(x)=determinant of the matrix xdiscrayabscondfloor(x)=floor of x=largest integer<=xfpnidealinv2ideallistunit(bnf,bound)=2-component vector [L,U] where L is as ideallistzstar, and U is a vector of vector of zinternallogs of the units, without generatorsidmatimagecomplincgam2(s,x)=incomplete gamma function (for debugging only)initellV=GGID3,L,pisdiagonalisprincipalraygen(bnf,x)=bnf being output by buchrayinit, gives [v,alpha,bitaccuracy], where v is the vector of exponents on the class group generators and alpha is the generator of the resulting principal ideal. In particular x is principal if and only if v is the zero vectorkerint(x)=LLL-reduced Z-basis of the kernel of the matrix x with integral entries using a modified LLLkerint2lindep2lllgramkerimlseriesellpolylog(m,x)=m-th polylogarithm of xpolylogd(m,x)=D_m~-modified m-th polylog of xpolzagrayclassnolist(bnf,liste)=if listes is as output by idealisunit or similar, gives list of corresponding ray class numbersrhoreal(x)=single reduction step of the binary quadratic form x of positive discriminantrnfdiscf(nf,pol)=given a pol with coefficients in nf, gives a 2-component vector [D,d], where D is the relative ideal discriminant, and d is the relative discriminant in nf^*/nf*^2srgcd(x,y)=polynomial gcd of x and y using the subresultant algorithmtaylor(x,y)=taylor expansion of x with respect to the main variable of ytwototwowf(raw)new_galois_formatnot a group in forsubgroupLLL_cmbf [no factor]gisirreducible
LLL_cmbf: %ld potential factors (tmax = %ld, bmin = %ld)
nextSousResultant j = %ld/%ldincorrect object in diagonalno more variables availableb8biddiffr2identifier (unknown code)not a suitable VECSMALL component#[%ld,%ld] please apply rnfequation(,,1)curve not defined over Rquadhilbertreal
Trying modulus = %Z and subgroup = %Z
E(4) = 2[x]2false nf in nf_get_signs = 0 is a pole (gzetakall)xbest = %Z
polredabs (precision problem)%020ld\cr}
den = real = \right) %s.gpunexpected endianness in %s[apell1] giant steps, i = %ldnot a prime in localredalgtobasis_ia not invertible in ff_PHlog_Fpczetapolynomial variable must have highest priority in nffactormodPrime ideal chosen: %Z
rowred j=%ldnewtonsums  (eq,er) = (%ld,%ld)
can't find symbol '%s' in dynamic symbol table of processplotcolor(w,c): in rectwindow w, set default color to c. Possible values for c are 1=black, 2=blue, 3=sienna, 4=red, 5=cornsilk, 6=grey, 7=gainsboroughpsdrawnot a module in nfsmithnon quadratic residue in gsqrtECM: time = %6ld ms,	ellfacteur giving up.
SQUFOF: ...found nothing on the ambiguous cycle
	after %ld steps there, time = %ld ms
SQUFOF: squfof_ambig returned %ld
`*where' out of bounds in ifac_sort_oneaprcl test fails! this is highly improbable[2]: mpqs_solve_linear_systemZ = %Z
delta[%ld] = %Z
d-1 test failed
S4GaloisConj:sigma=%Z
GaloisConj: %d hops on %Z tests
      Working with polynomial #%ld:
not a t_QFR in powrealrawlllint[2], kmax = %ld
...LLL reducing precision to %ld
Matrix to be reduced:
%Z
lllgramallgenparametersnot a vector in forvecnot a vector of two-component vectors in forvecnon integral index in prodinfprime_loop_init  Baker = %Z
Internal error in sv2pari!Math::Pari::PARIMath::Pari::interface14Math::Pari::interface23$$;$Math::Pari::interface28$$$$;$Access to elements of not-a-vector_gbitneg_gneg_log%s not implemented on this architecturetag%li items moved off stack, onStack=%ld, offStack=%lda, b, c=0rootpadic
  current stack size: %lu (%.3f Mbytes)
Eulervrrallocatemem({s=0}): allocates a new stack of s bytes. doubles the stack if s is omittedasin(x): inverse sine of xbesselibezoutres(x,y): gives a 3-dimensional row vector [u,v,d] such that d=resultant(x,y) and u*x+v*y=d, where x and y are polynomialsbitxor(x,y): bitwise "exclusive or" of two integers x and y. Negative numbers behave as if modulo big power of 2bnrL1bnrclassnocorediscelllseries(e,s,{A=1}): L-series at s of the elliptic curve e, where A a cut-off point close to 1GD&ellrootnoforidealappridealintersectidealmin(nf,ix,{vdir}): minimum of the ideal ix in the direction vdir in the number field nfidealprimedec(nf,p): prime ideal decomposition of the prime number p in the number field nf as a vector of 5 component vectors [p,a,e,f,b] representing the prime ideals pZ_K+a. Z_K, e,f as usual, a as vector of components on the integral basis, b Lenstra's constantintfuncinit(X=a,b,s,{flag=0},{m=0}): initialize tables for integrations  from a to b using a weight s(X). Essential for integral transforms such as intmellininv, intlaplaceinv and intfourier, since it avoids recomputing all the time the same quantities. Must then be used with intmellininvshort (for intmellininv) and directly with intnum and not with the corresponding  integral transforms for the others. See help for intnum for coding of a  and b, and m is as in intnuminit. If flag is nonzero, assumes that  s(-X)=conj(s(X)), which is twice faster.lpkillmatalgtobasismatintersect(x,y): intersection of the vector spaces whose bases are the columns of x and ymatsupplementmattranspose(x): x~=transpose of xnextnfeltmodnfgaloisapplynfisidealDGGpolredpolresultantpolsubcycloLGD0,L,pprintp1qfjacobi(x): eigenvalues and orthogonal matrix of eigenvectors of the real symmetric matrix xqfperfection(a): rank of matrix of xx~ for x minimal vectors of a gram matrix aquadclassunitrnfdedekind(nf,T,pr): relative Dedekind criterion over nf, applied to the order defined by a root of irreducible polynomial T, modulo the prime ideal pr. Returns [flag,basis,val], where basis is a pseudo-basis of the enlarged order, flag is 1 iff this order is pr-maximal, and val is the valuation in pr of the order discriminantrnfeltabstorelrnfeltup(rnf,x): expresses x (belonging to the base field) on the relative fieldserconvolsin(x): sine of xsinhsubst(x,y,z): in expression x, replace the variable y by the expression ztracevaluationznlogargument too large in ggammasum from 0 to N-1R1 = %ld, R2 = %ld
D = %Z
 %ldsmall_norm (precision too low)BOUND = %.4g
agm(x,y)=arithmetic-geometric mean of x and yGLLpanelldiscfeint1(x)=exponential integral E1(x)factor(x)=factorization of xfactorpadic(x,p,r)=p-adic factorization of the polynomial x to precision r, using the round 4 algorithmforstep(X=a,b,s,seq)=the sequence is evaluated, X going from a to b in steps of sglobalredideallistunitarch(bnf,lists,arch)=adds the archimedean arch to the lists output by ideallistunitinitalgred(x)=x being a nonconstant irreducible polynomial, finds (using polred) a simpler polynomial pol defining the same number field, and gives the vector: [pol,[r1,r2],discf,index,[M,MC,T2,T,different] (see manual), r1+r2 first roots, integral basis, matrix of power basis in terms of integral basis, multiplication table of basis]isinclkbessel(nu,x)=K-bessel function of index nu and argument x (x positive real of type real, nu of any scalar type)nfhermitenorm(x)=norm of xo(a^b)=O(a^b)=p-adic or power series zero with precision given by bphi(x)=Euler's totient function of xsetintersect(x,y)=intersection of the sets x and ysize(x)=maximum number of decimal digits minus one of (the coefficients of) xsqred(x)=square reduction of the (symmetric) matrix x ( returns a square matrix whose i-th diagonal term is the coefficient of the i-th square in which the coefficient of the i-th variable is 1)weberf2(x)=Weber's f2 function of x (j=(f2^24+16)^3/f2^24)(on with colors)prettyprinteryesFp_sqrtnQXQ_inv: char 0 check failedlast factor still to be checked
Root boundcan't deflate this power series (d = %ld): %Zfactor of general polynomialQ_divmuli_to_intTneed exponential decrease in intinvmellinshortboth nonzero real and imag. part in coding, real ignoredgauss_pivot_ker. k=%ld, n=%ldnot a valid identifierfthis should be an integerbuchquadincorrect parameters in quadclassunitsubFBquad (%ld elt.)A3S4you found a counter-example to a conjecture, please report!t_MATembedded braces (in parser)Qfb(#<%d>goodsingular curve in ellinitmodule too large in Fp_shanksveceint1k < 0 in thetanullkpolynomial variable must have highest priority in nffactor with parameters: %Z^%ld
plotplotmove(w,x,y): move cursor to position x,y in rectwindow wlarge Minkowski bound: certification will be VERY long%9.*gECM: time = %6ld ms, entering B2 phase, p = %lu
ECM: finishing curves %ld...%ld
SQUFOF: entering main loop with forms
	(1, %ld, %ld) and (1, %ld, %ld)
	of discriminants
	%Z and %Z, respectively
IFAC: checking for odd power
IFAC: trying Lenstra-Montgomery ECM
MPQS: starting main loop
MPQS: kN has %ld decimal digits
,<%lu>MPQS: sizing out of tune, FB size or tolerance
	too largefrobenius powerGaloisConj:I will try %Z permutations
hnf[2]. i=%lddep = %Z
Terminal size directive without ','partition functiongramtm1 = %ZlllintpartialallpslqL2all_roots: restarting, i = %ld, e = %ld
forbidden or incompatible types in hilgisprimeintegral part not significant in sfcontnegative number of components in vector* real root no %ld/%ld
Semirat. reduction: B0 -> %Z x <= %Z
LLL_First_Pass successful !!
D0,G,$$Math::Pari::interface18Math::Pari::interface32Math::Pari::pari_pprintMath::Pari::changevalue$Math::Pari::initmem not defined!Same iterator for a double loopgle_gbitxor_gltLGD0,Lvalzero polynomial in FpXQ_pow. %Z not primeQpXQ_to_ZXYmallocing NULL objectbot=0x%lx	top=0x%lx
Piallocatemembernvec(x): Vector of rational Bernoulli numbers B_0, B_2,...up to B_(2x)bnfinit(P,{flag=0},{tech=[]}): compute the necessary data for future use in ideal and unit group computations, including fundamental units if they are not too large. flag and tech are both optional. flag can be any of 0: default, 1: insist on having fundamental units, 2: do not compute units, 3: small bnfinit, which can be converted to a big one using bnfmake. See manual for details about techbnfisprincipal(bnf,x,{flag=1}): bnf being output by bnfinit (with flag<=2), gives [v,alpha], where v is the vector of exponents on the class group generators and alpha is the generator of the resulting principal ideal. In particular x is principal if and only if v is the zero vector. flag is optional, whose meaning is: 0: output only v; 1: default; 2: output only v, precision being doubled until the result is obtained; 3: as 2 but output generatorsGDGD0,L,pbnrclassbnrisprincipal(bnr,x,{flag=1}): bnr being output by bnrinit, gives [v,alpha], where v is the vector of exponents on the class group generators and alpha is the generator of the resulting principal ideal. In particular x is principal if and only if v is the zero vector. If (optional) flag is set to 0, output only vcharpolychinese(x,{y}): x,y being both intmods (or polmods) computes z in the same residue classes as x and ycontfracpnqn(x): [p_n,p_{n-1}; q_n,q_{n-1}] corresponding to the continued fraction xelladdGGD2,L,pelllocalred(e,p): e being an elliptic curve, returns [f,kod,[u,r,s,t],c], where f is the conductor's exponent, kod is the Kodaira type for e at p, [u,r,s,t] is the change of variable needed to make e minimal at p, and c is the local Tamagawa number c_pfactorintidealprimedecissquarelistput(list,x,{n}): sets n-th element of list equal to x. If n is omitted or greater than the current list length, just append xmathnfmatrank(x): rank of the matrix xmin(x,y): minimum of x and ynfeltpowmodpr(nf,a,k,pr): element a^k modulo pr in nf, where pr is in modpr format (see nfmodprinit)numbpart(x): number of partitions of xpolredord(x): reduction of the polynomial x, staying in the same orderprimepiprodqfperfectionrnfidealdown(rnf,x): finds the intersection of the ideal x with the base fieldrnfisnorminitrnfpolredabsshiftmulsubstvec(x,v,w): in expression x, make a best effort to replace the variables v1,...,vn by the expression w1,...,wnzetakinitzncoppersmithbuchcertifybuchrayinitgenclassnodecodefactorhilbpiideallistzstargenlindep2(x,dec)=Z-linear dependencies between components of x using LLL, where dec should be about one half the number of decimal digits of precisionlogagmmatrixqz3(x)=finds a basis of the intersection with Z^n of the Q-vector space spanned by the columns of xminimreduceddisc(f)=vector of elementary divisors of Z[a]/f'(a)Z[a], where a is a root of the polynomial frootsof1(nf)=number of roots of unity and primitive root of unity in the number field nfvecindexsortStep 3
beta LLL-reduced mod U^l = %Z
   datadir = "%s"
comment>    %s = 1 (on)
bitwise negationZY_ZXY_resultant_all: LERS needs lambdaexact type in subgrouplistfor this typeprod: remaining objects %ld
x = 0 in Fourierqrom3: iteration %ld: %Z
FpM_gauss. i=%ldFlxqM_kerUnrecognized id '%s' in a stringified flagclgpzkstthis function uses a killed variableynot a set in setminuschk_gen_init: skipfirst = %ld
[pipe:] '%s' failed[&=%016lx] a transcendental function%Z should divide valuation (= %ld) in sqrtnser_powk not a positive even integer in elleisnumcut-off point must be positive in lseriesellinitell for 2-adic numbersCM_ellpowforbidden substitution in a scalar typezero argument in O()main variable must have higher priority in gtosergreal/gimagtab[q^-s] from 1 to N-1cannot give a power series result in k/n bessel functionnon-positive valuation in etanfsqffResult for prime %Z is:
%Z
not a factorisation in nfbasisinstallvrrD"",r,D"",s,vV=GGIDGDGpplotboxvLGGnot a correct ideal list in %sget_archold method: y = %Z, M0 = %Z
Searching minimum of T2-form on units:
valuation (valp) overflowzero polynomialbad argument for an elliptic curve related function  for one roundifac_startIFAC: main loop: repeated new factor
	%Z
	%Z
ifac_crack [Z_issquarerem miss]ifac_sort_onecomposite equals prime in ifac_sort_one
MPQS: passing the %3.1f%% sort point
 and combining\\ KERNEL COMPUTED BY MPQS
KERNEL=shorter
Subfields of degree %ld: %Z
embedding = %Z
MonomorphismLift: true early solution.
f=%Z
 borne=%Z
 l-borne=%Z
S4GaloisConj:Testing %d/3:%d/2:%d/2:%d/4:%Z:%Z
GaloisIndex: Using hash value u=%ld
incompatible field degrees in conjvec[2]: smithall, i = %ldarch/
Checking LLL basis
Increasing dim, delta = %d t = %d
primitive root mod %Z does not existnegative argument in factorial functionMath::Pari::STOREMath::Pari::interface199$;$Math::Pari::interface87Math::Pari::_2boolMath::Pari::DESTROYname, v = 99bernzonebad index in listinsertFpX_factor_2new bloc, size %6lu (no %ld): %08lx
-->Polrev(x,{v=x}): convert x (usually a vector or a power series) into a polynomial with variable v, starting with the constant termVecsmallVecsmall({x=[]}): transforms the object x into a VECSMALL. Empty vector if x is omittedacosh(x): inverse hyperbolic cosine of xbnfclassunit(P,{flag=0},{tech=[]}): compute the class group, regulator of the number field defined by the polynomial P, and also the fundamental units if they are not too large. flag and tech are both optional. flag can be any of 0: default, 1: insist on having fundamental units, 2: do not compute units. See manual for details about tech. P may also be a non-zero integer, and is then considered as the discriminant of a quadratic orderlGDGDGV=GGEDGellpointtoz(e,P): lattice point z corresponding to the point P on the elliptic curve eellsigma(om,z,{flag=0}): om=[om1,om2], value of the Weierstrass sigma function of the lattice generated by om at z if flag = 0 (default). If flag = 1, arbitrary determination of the logarithm of sigma. If flag = 2 or 3, same but using the product expansion instead of theta serieselltaniyama(e): modular parametrization of elliptic curve efactornffibonacci(x): fibonacci number of index x (x C-integer)hilbertidealaddtoone(nf,x,{y}): if y is omitted, when the sum of the ideals in the number field K defined by nf and given in the vector x is equal to Z_K, gives a vector of elements of the corresponding ideals who sum to 1. Otherwise, x and y are ideals, and if they sum up to 1, find one element in each of them such that the sum is 1idealred(nf,x,{vdir=0}): LLL reduction of the ideal x in the number field nf along direction vdir, in HNFidealvalintmellininvnfisideal(nf,x): true(1) if x is an ideal in the number field nf, false(0) if notprecprimeprinttex(a): outputs a in TeX formatqfbnucompqflllgramqflllgram(x,{flag=0}): LLL reduction of the lattice whose gram matrix is x (gives the unimodular transformation matrix). flag is optional and can be 0: default,1: lllgramint algorithm for integer matrices, 4: lllgramkerim giving the kernel and the LLL reduced image, 5: lllgramkerimgen same when the matrix has polynomial coefficients, 8: lllgramgen, same as qflllgram when the coefficients are polynomialsrnfdet(nf,order): given a pseudomatrix, compute its pseudodeterminantrnfeltreltoabs(rnf,x): transforms the element x from relative to absolute representationrnfidealabstorel(rnf,x): transforms the ideal x from absolute to relative representationrnfkummerserlaplace(x): replaces the power series sum of a_n*x^n/n! by sum of a_n*x^n. For the reverse operation, use serconvol(x,exp(X))lGGD0,L,znstargpsi(jid=%ld,jdir=%ld)subcycloadj(x)=adjoint matrix of xatan(x)=inverse tangent of xatanh(x)=inverse hyperbolic tangent of xbasisbasis2(x)=integral basis of the field Q[a], where a is a root of the polynomial x, using the round 2 algorithmbin(x,y)=binomial coefficient x*(x-1)...*(x-y+1)/y! defined for y in Z and any xcfcvtoi(x)=truncation of x, without taking into account loss of integer part precisiondetint(x)=some multiple of the determinant of the lattice generated by the columns of x (0 if not of maximal rank). Useful with hermitemodgaloisapplygaloisconjforce(nf)=list of conjugates of a root of the polynomial x=nf[1] in the Galois number field nf, using p-adics, LLL on integral basis. Guaranteed to be complete if the field is Galois, otherwise there is an infinite loopideallistunitgenisinclfast(nf1,nf2)=tests whether the number nf1 is isomorphic to a subfield of nf2 or not. If it gives a non-zero result, this proves that this is the case. However if it gives zero, nf1 may still be isomorphic to a subfield of nf2 so you have to use the much slower isincl to be sureisirreducible(x)=true(1) if x is an irreducible non-constant polynomial, false(0) if x is reducible or constantisoncurve(e,x)=true(1) if x is on elliptic curve e, false(0) if notisset(x)=true(1) if x is a set (row vector with strictly increasing entries), false(0) if notlexsortmat(x)=transforms any GEN x into a matrixnupowpowell(e,x,n)=n times the point x on elliptic curve e (n in Z)rnfequation2(nf,pol)=given a pol with coefficients in nf, gives [apol,th], where apol is the absolute equation of the number field defined by pol and th expresses the root of nf[1] in terms of the root of apolrnfisfree(bnf,order)=given an order as output by rnfpseudobasis or rnfsteinitz, outputs true (1) or false (0) according to whether the order is free or notsortthreetotwo2(nf,a,b,c)=returns a 3-component vector [d,e,U] such that U is a unimodular 3x3 matrix with algebraic integer coefficients such that [a,b,c]*U=[0,d,e][rnfkummer] candidate list(use old functions, don't ignore case)parisizenoneGPHELPTrying lambda = %ld
can't factor %Zsrgcd: dr = %ld
shallowtranshnf_invimage[install] updating '%s' prototype; module not reloadedbreak not allowed after !break not allowed here (expanding string)global variable not allowedreal_relations#### Tentative regulator: %Z
incompatible variables in gredCompute %sstark (computation impossible)S1galois (bug3)get_bnfpolmult. tableget_red_Gchk_gen_init: generator %Z
%3ld:%3ld ]

 / close0 argument in cotanXcan't evaluate ellzeta at a pole%lu is not prime, use ellakinvalid quadratic form in qfevalgtoserq >= 1 in thetazero argument in a k/n bessel functionGroup(())IndexPartial: factor %Z^%ld plotrbox(w,dx,dy): if the cursor is at (x1,y1), draw a box with diagonal (x1,y1)-(x1+dx,y1+dy) in rectwindow w (cursor does not move)not a matrix in nfsmith[1]: discrayabslistarchneither bnf nor bnr in conductor or discraymulti-curves cannot be plot recursivelyifac_sumdivStep4: q-values (# = %ld, largest = %ld): MPQS: Ran out of primes for A, giving up.
MPQS: chose Q_%ld(x) = %Z x^2 %c %Z x + C
MPQS: chose primes for AMonomorphismLift: false early solution.
GaloisConj:Entree Init Test
S4GaloisConj:pj=%Z
Combinatorics too hard : would need %Z tests!
 I'll skip it but you will get a partial result...hnf_special    matb cleaned up (using Id block)
hnfspec [%ld x %ld] --> [%ld x %ld]non coprime ideals in hnfmergeGaloisbig: reduced polynomial #1 = %Z

*** Entering isin_%ld_G_H_(%ld,%ld)
gnuplot-like plotting environment not loaded yetCan't resolve 'get_term_ftable' function from Gnuplot drawing engine '%s': %sdiscriminant not congruent to 0,1 mod 4 in %sdifferent discriminants in qfb_compnot a t_REAL in 4th component of a t_QFRinvalid polynomial in thue (need n>2)B0 -> %Z
Math::Pari::sv2pariMath::Pari::int_set_term_ftableg, eltCorrupted data: should be variablegadd_gge_gmodgabsgaffect (gen_0)gaffect (pol_1/pol_x)polfnfFqX_factorStrtex({str}*): translates its (string) arguments to TeX format and returns the resulting stringacosbestapprbnfissunitbnfsunit(bnf,S): compute the fundamental S-units of the number field bnf output by bnfinit, S being a list of prime ideals. res[1] contains the S-units, res[5] the S-classgroup. See manual for detailsbnrclass(bnf,ideal,{flag=0}): given a big number field as output by bnfinit (only) and an ideal or a 2-component row vector formed by an ideal and a list of R1 zeros or ones representing a module, finds the ray class group structure corresponding to this module. flag is optional, and can be 0: default, 1: compute data necessary for working in the ray class group, for example with functions such as bnrisprincipal or bnrdisc, without computing the generators of the ray class group, or 2: with the generators. When flag=1 or 2, the fifth component is the ray class group structure obtained when flag=0bnrdisclistconcatconcat(x,{y}): concatenation of x and y, which can be scalars, vectors or matrices, or lists (in this last case, both x and y have to be lists). If y is omitted, x has to be a list or row vector and its elements are concatenatedGGGpellheightvGVIgaloisfixedfield(gal,perm,{flag},{v=y}): gal being a galois field as output by galoisinit and perm an element of gal.group or a vector of such elements, return [P,x] such that P is a polynomial defining the fixed field of gal[1] by the subgroup generated by perm, and x is a root of P in gal expressed as a polmod in gal.pol. If flag is 1 return only P. If flag is 2 return [P,x,F] where F is the factorization of gal.pol over the field defined by P, where the variable v stands for a root of Pgaloisisabelian(gal,{flag=0}): gal being as output by galoisinit, return 0 if gal is not abelian, the HNF matrix of gal over gal.gen if flag=0, 1 if flag is 1, and the SNF of gal is flag=2idealmul(nf,x,y,{flag=0}): product of the two ideals x and y in the number field nf. If (optional) flag is non-nul, reduce the resultidealtwoelt(nf,x,{a}): two-element representation of an ideal x in the number field nf. If (optional) a is non-zero, first element will be equal to alGGGincgamc(s,x): complementary incomplete gamma functionVGGED0,L,D0,L,pmatdetmatisdiagonal(x): true(1) if x is a diagonal matrix, false(0) otherwisematmultodiagonal(x,y): product of matrices x and y, knowing that the result will be a diagonal matrix. Much faster than general multiplication in that casematpascalmatrixmatsolve(M,B): gaussian solution of MX=B (M matrix, B column vector)nffactornffactormod(nf,pol,pr): factorize polynomial pol modulo prime ideal pr in number field nfpolinterpolatepolylog(m,x,{flag=0}): m-th polylogarithm of x. flag is optional, and can be 0: default, 1: D_m~-modified m-th polylog of x, 2: D_m-modified m-th polylog of x, 3: P_m-modified m-th polylog of xqfbclassnoqfbhclassno(x): Hurwitz-Kronecker class number of x>0qfsignquadpolytan(x): tangent of xtaylor(x,y): taylor expansion of x with respect to the main variable of yteichmuller(x): teichmuller character of p-adic number xvecextract(x,y,{z}): extraction of the components of the matrix or vector x according to y and z. If z is omitted, y designs columns, otherwise y corresponds to rows and z to columns. y and z can be vectors (of indices), strings (indicating ranges as in "1..10") or masks (integers whose binary representation indicates the indices to extract, from left to right 1, 2, 4, 8, etc.)writezetak(nfz,s,{flag=0}): Dedekind zeta function of the number field nfz at s, where nfz is the vector computed by zetakinit (NOT by nfinit) flag is optional, and can be 0: default, compute zetak, or non-zero: compute the lambdak function, i.e. with the gamma factorsglngammazero ideal in isprincipalSubcyclo: %ld orbits with %ld elements each
char2cyclo(n)=n-th cyclotomic polynomialdilog(x)=dilogarithm of xdiscGD0,G,D0,G,D0,L,eigenfactcantor(x,p)=factorization mod p of the polynomial x using Cantor-Zassenhausfactoredbasisfactorpadic2galoisconj1(nf)=list of conjugates of a root of the polynomial x=nf[1] in the same number field nf, using complex numbers, LLL on integral basis (not always complete)hclassno(x)=Hurwitz-Kronecker class number of x>0hilbert(n)=Hilbert matrix of order n (n C-integer)initalgred2(P)=P being a nonconstant irreducible polynomial, gives a two-element vector [nf,mod(a,pol)], where nf is as output by initalgred and mod(a,pol) is a polymod equal to mod(x,P) and pol=nf[1]initzeta(x)=compute number field information necessary to use zetak, where x is an irreducible polynomialisisomfast(nf1,nf2)=tests whether the number fields nf1 and nf2 are isomorphic or not. If it gives a non-zero result, this proves that they are isomorphic. However if it gives zero, nf1 and nf2 may still be isomorphic so you have to use the much slower isisom to be surejacobikbessel2lllintlllkerim(x)=kernel and lll reduction of the vectors forming the integral matrix xlllkerimgen(x)=kernel and lll reduction of the vectors forming the matrix x with polynomial coefficientslocalredmathellmatrix(m,n,X,Y,expr)=mXn matrix of expression expr, the row variable X going  from 1 to m and the column variable Y going from 1 to nmatsize(x)=number of rows and columns of the vector/matrix x as a 2-vectormu(x)=Moebius function of xnfdivnfhermitemodnfmodnumdiv(x)=number of divisors of xpadicprec(x,p)=absolute p-adic precision of object xpointell(e,z)=coordinates of point on the curve e corresponding to the complex number zpolredabsnored(x)=a smallest generating polynomial of the number field for the T2 norm on the roots without initial polredpolylogdpolylogdoldsumpos2(X=a,expr)=sum of positive series expr, the formal variable X starting at a, using Zagier's polynomialsunit(x)=fundamental unit of the quadratic field of discriminant x where x must be positivepolred(compositum) = %Z
get_sep: argument too long (< %ld chars)arguments must be positive integers%c%ld.%ldechoPARIbreak\hskip 0pt plus \hsize\relax\discretionary{}{}{}}   path = "%s"
RgX_extgcd, dr = %ldnot a squarefree polynomial in sturmdet. col = %ldNon-numeric argument of an action in a templatecycexpr%s[1-%ld]negative index in vecsortelement is not in the base field in rnfelementdown	character no: %ld (%ld/%ld)
polynomial not in Z[X] in %sS3A_4(6) = [2^2]3 Occupation of the PARI stack : %6.2f percent
%016lx  gpow: 0 to a forbidden powergexpincompatible p-adic numbers in initellnorm too large in CMnot a variable in substvecincorrect object in O()for this exponent, GSmin = %Z
Time reduction: %ld
get_normrnfordmaxzpsolublenfplotrmove(w,dx,dy): move cursor to position (dx,dy) relative to the present position in the rectwindow wnot a correct ideal list in nfsmithincorrect subgroup in %ssnextpr: %lu should have been prime but isn't
ellfacteursIFAC: trying Pollard-Brent rho method
Step6: testing potential divisors
MPQS: factoring number of %ld decimal digits
MPQS: size of factor base = %ld
MPQS: found %ld full relations
MPQS:            %4.3g full relations per 100 polynomials
manyMPQS: FB [-1,2overflow in calc_blockGaloisConj:increase prec of p-adic roots of %ld.
GaloisConj:Testing S4 first
GaloisConj:Orbite:%Z
GaloisConj:Fini!
A4GaloisConj:%ld hop sur %d iterations max
Combinatorics too hard : would need %Z tests!
I will skip it, but it may induce an infinite loopmore rows than columns in matrixqzhnflll-perl -MDynaLoader -we %cpackage DynaLoader; print mod2fname([qw(Term Gnuplot)]) if defined &mod2fname%cnegative definite t_QFIfirst LLL: prec = %ld
minim0, rank>=%ldFlx_to_Flv  c15 = %Z
ratilft: amax must be >= 0, found
	amax=%Z
Math::Pari::pari2num_;@Math::Pari::interface11Math::Pari::interface209Math::Pari::interface33Math::Pari::pari_version_expMath::Pari::__detach_stackXSUB call through interface with a NULL codearg1, arg2=0_gcmp0_abscvtop2normalizegsignefactmoddoubling stack size; new stack = %lu (%.3f Mbytes)I=I(): square root of -1VecvD0,L,bernfrac(x): Bernoulli number B_x, as a rational numberbesselnbestappr(x,k): gives the best approximation to the real x with denominator less or equal to kbezout(x,y): gives a 3-dimensional row vector [u,v,d] such that d=gcd(x,y) and u*x+v*y=dbnfclgpbnfregbnrclassno(bnf,x): ray class number of the module x for the big number field bnf. Faster than bnrclass if only the ray class number is wantedGDGDGDGcontentcontfrac(x,{b},{lmax}): continued fraction expansion of x (x rational,real or rational function). b and lmax are both optional, where b is the vector of numerators of the continued fraction, and lmax is a bound for the number of terms in the continued fraction expansiondefault({opt},{v}): returns the current value of the current default opt. If v is present, set opt to v first. If no argument is given, print a list of all defaults as well as their values.divisorseint1(x,{n}): exponential integral E1(x). If n is present, computes the vector of the first n values of the exponential integral E1(n.x) (x > 0)elleisnumellheight(e,x,{flag=2}): canonical height of point x on elliptic curve E defined by the vector e. flag is optional and selects the algorithm used to compute the archimedean local height. Its meaning is 0: use theta-functions, 1: use Tate's method, 2: use Mestre's AGMellordinate(e,x): y-coordinates corresponding to x-ordinate x on elliptic curve eellztopoint(e,z): coordinates of point P on the curve e corresponding to the complex number zidealnorm(nf,x): norm of the ideal x in the number field nfimag(x): imaginary part of xintmellininvshort(sig,x,tab): numerical integration on the  line real(z) = sig (or sig[1]) of s(z)x^(-z)dz/(2*I*Pi), i.e. inverse Mellin transform of s at x. sig is coded as follows: either it is real, and then by default assume s(z) decreases like exp(-z). Or sig = [sigR, al], sigR is the abcissa of integration, and al = 0 for slowly decreasing functions, or al > 0 if s(z) decreases like exp(-al*z). Compulsory table tab has been  precomputed using the command intfuncinit(t=[[-1],sig[2]],[[1],sig[2]],s)  (with possibly its two optional additional parameters), where sig[2] = 1 if not given. Orders of magnitude faster than intmellininv.intnumintnuminit(a,b,{m=0}): initialize tables for integrations from a to b. See help for intnum for coding of a and b. Possible types: compact interval, semi-compact (one extremity at + or - infinity) or R, and very slowly, slowly or exponentially decreasing, or sine or cosine oscillating at infinities,listcreate(n): creates an empty list of maximum length nmatdiagonalmatmuldiagonalnextprimenfhilbert(nf,a,b,{p}): if p is omitted, global Hilbert symbol (a,b) in nf, that is 1 if X^2-aY^2-bZ^2 has a non-trivial solution (X,Y,Z) in nf, -1 otherwise. Otherwise compute the local symbol modulo the prime ideal pnfisinclnumeratorpolrootsqfbpowraw(x,n): n-th power without reduction of the binary quadratic form xrnfidealabstorelsetsearch(x,y,{flag=0}): looks if y belongs to the set x. If flag is 0 or omitted, returns 0 if it is not, otherwise returns the index j such that y==x[j]. If flag is non-zero, return 0 if y belongs to x, otherwise the index j where it should be insertedsqrtintGnPthetavaluation(x,p): valuation of x with respect to pzetared_mod_units
#### Tentative class number: %Z
++ LV[%ld] = %Z  nb. fact./nb. small norm = %ld/%ld = %.3f
SubCyclo: testing %ld^%ld
Subcyclo: complex=%ld
O(a^b)=o(a^b)=p-adic or power series zero with precision given by bbuchnarrow(bnf)=given a big number field as output by buchinitxx, gives as a 3-component vector the structure of the narrow class groupGnconductor(bnr,subgroup)=conductor of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroupdegree(x)=degree of the polynomial or rational function x. -1 if equal 0, 0 if non-zero scalardiscrayabslistarchdiscrayabslistarch(bnf,arch,bound)=gives list of discrayabscond of all modules up to norm bound with archimedean places arch, in a longvector formaterfc(x)=complementary error functiongaloisimage(x)=basis of the image of the matrix xisidealkbessel2(nu,x)=K-bessel function of index nu and argument x (x positive real of type real, nu of any scalar type)krolll1lllgenomega(x)=number of unrepeated prime divisors of xpascalpolredabsallprodinf(X=a,expr)=infinite product (X goes from a to infinity) of real or complex expressionquadgen(x)=standard generator of quadratic order of discriminant xseriessupplement(x)=supplement the columns of the matrix x to an invertible matrixveclexsortzidealstarinit   realprecision = %ld significant digits(use old functions, ignore case)   %s = 0 (off)
    mu'= not a positive exponent in polhenselliftnot a factorisation in factorbackdet, col %ld / %ldimpossible concatenation: %s %Z . %s %ZFqM_gauss_pivotnumeric id in a stringified flagareac4; or ] expected %ldPnot a series in laplacemain variable must be of higher priority in rnfinitalgTrying to find another modulus...diff(CHI) = %Z* Root Number: cond. no %ld/%ld (%ld chars)
false nf in nf_get_r1
initzeta: N0 = %Z
%c[%ld;%ld;%ldmt_QFRNULLList([mat(deletegloganell for n >= %lunot a complex multiplication in powelltorsell (bug2)zsignegdivmodgsubstinvalid base change matrix in qf_base_changegzetagpolylogzmodpr initialized for integers only!0 argument in nfhilbertpplease apply rnfisnorminit firstplotdrawincorrect character length in KerCharunknown function or error in formal parametersvariable name expectedobsolete functionconstant polynomialnot a vector in rectdrawPL: N-1 factored!
Rho: time = %6ld ms,	%3ld rounds, back to normal mode
IFAC: main loop: repeated old factor
	%Z
IFAC: after main loop: repeated old factor
	%Z
aprcl: e(t) too small 1 1p = %Z, lift to p^%ld
FixedField: Weight: %Z
    after phase2:
hnflll (reducing), i = %ldhnfall[3], j = %ldgalois in degree > 11square discriminant in %sd must be positivelllfp giving upincrementalGSgenbase change matrix =
fundunitinsufficient precision for p = 2 in hilbertlist of numerators too short in sfcontf2    B0 -> %Z
looking for a fundamental unit of norm -1
ratlift: must have 0 <= x < m, found
	x=%Z
	m=%Z
Math::PariAttempt to ask Perl to free PARI function not installed from PerlMath::Pari::sv2parimatMath::Pari::interface299$$$$$;$$Math::Pari::reset_on_reloadgcmp_1_gbitand_geqlGGvV=GGvLGGGmatsizecomparisongaffect (gpi)normalizing a polynomial with 0 leading termrootsoldGP_DATA_DIRtypeerinvmoderList({x=[]}): transforms the vector or list x into a list. Empty list if x is omittedGGGDGpargbesseljhGLGD-1,L,ellchangepointvs*exp(x): exponential of xfactor(x,{lim}): factorization of x. lim is optional and can be set whenever x is of (possibly recursive) rational type. If lim is set return partial factorization, using primes up to lim (up to primelimit if lim=0)galoisfixedfieldidealinvideallistarch(nf,list,arch): list is a vector of vectors of of bid's as output by ideallist. Return a vector of vectors with the same number of components as the original list. The leaves give information about moduli whose finite part is as in original list, in the same order, and archimedean part is now arch. The information contained is of the same kind as was present in the input.matimagecompl(x): vector of column indices not corresponding to the indices given by the function matimagematranknfdetintnfhnf(nf,x): if x=[A,I], gives a pseudo-basis of the module sum A_jI_jnfisisom(x,y): as nfisincl but tests whether x is isomorphic to ynfsnf(nf,x): if x=[A,I,J], outputs [c_1,...c_n] Smith normal form of xGGGLpolisirreducible(x): true(1) if x is an irreducible non-constant polynomial, false(0) if x is reducible or constantprintp1(a): outputs a (in beautified format) without ending with newlineqfbprimeform(x,p): returns the prime form of discriminant x, whose first coefficient is pqfbred(x,{flag=0},{D},{isqrtD},{sqrtD}): reduction of the binary quadratic form x. All other args. are optional. D, isqrtD and sqrtD, if present, supply the values of the discriminant, floor(sqrt(D)) and sqrt(D) respectively. If D<0, its value is not used and all references to Shanks's distance hereafter are meaningless. flag can be any of 0: default, uses Shanks's distance function d; 1: use d, do a single reduction step; 2: do not use d; 3: do not use d, single reduction step.quadregulator(x): regulator of the real quadratic field of discriminant xrealreturnrnfidealmul(rnf,x,y): relative version of idealmul, where rnf is a relative numberfieldsetrandGnGtrapvecmaxvecsortKCZ = %ld, KC = %ld, n = %ld
*subcyclo_rootsaddell(e,z1,z2)=sum of the points z1 and z2 on elliptic curve eakellbasistoalgbuchgenforcefucf(x)=continued fraction expansion of x (x rational,real or rational function)char2(x,y)=characteristic polynomial of the matrix x expressed with variable y, using the Hessenberg form. Can be much faster or much slower than char, depending on the base ringclassno2compo(x,s)=the s'th component of the internal representation of x. For vectors or matrices, it is simpler to use x[]content(x)=gcd of all the components of x, when this makes senseconvol(x,y)=convolution (or Hadamard product) of two power seriesdivsumfactoredpolred2(x,p)=reduction of the polynomial x, where p is the matrix of the factorization of the discriminant of x (gives elements and minimal polynomials)hermitepermhermiteperm(x)=3-component vector [H,U,P] such that H is an (upper triangular) Hermite normal form of x with extra zero columns, U is a unimodular matrix and P is a permutation of the rows such that P applied to xU gives H, using Batut's algorithmincgam4(s,x,y)=incomplete gamma function where y=gamma(s) is precomputedV=GGID0,L,pisprincipalgenkerint1powellpowrealraw(x,n)=n-th power without reduction of the binary quadratic form x of positive discriminantprec(x,n)=change the precision of x to be n (n C-integer)principalidealrandom()=random integer between 0 and 2^31-1reverseset(x)=convert x into a set, i.e. a row vector with strictly increasing coefficientsshift(x,n)=shift x left n bits if n>=0, right -n bits if n<0shiftmul(x,n)=multiply x by 2^n (n>=0 or n<0)trunc(x)=truncation of x;when x is a power series,take away the O(X^)vvector(n,X,expr)=column vector with n components of expression expr (X ranges from 1 to n)zidealstarinitgenmain variable in kummer must not be xbug%d in kummer(no backward compatibility)   prompt%s = "%s"
non invertible polynomial in FpXQ_invnb subgroup = %ld
missing case in gdivexactinfinities of the same sign in intnuminitgentrying to concat elements of an empty vectorcodifftoo many parameters in user-defined function callsmith/class groupnot vectors in polinterpolateIt's not a square...
RecCoeff (cf = %ld, B = %Z)
polynomial not in Z[X,Y] in %sfalse nf in nf_get_r2matrix M\cr
 I/O: can't remove file %s%s is set (%s), but is not writeable(%c,lgefint=%ld): %s .%ld.%ld  z1 = %Z
reduction mod SL2 (reduce_z)\a log/atan appears in intformalgrndtoiinvalid vector in qfbevalEntering veceint1:
szetaargument must belong to upper half-planeTreating p^k = %Z^%ld
this combination of flags in rnfpolredabsnewtonsums: result doesn't fit in cache
main variable must be of higher priority in rnfisnorminitnot a factorization in idealapprfactLbnrclassnonot a definite matrix in lllgramflag PLOT_PARAMETRIC ignoredsnextpr: integer wraparound after prime %lu
MPQS: found %3.1f%% of the required relations
,%lu...] Wait a second --
MPQS: computing logarithm approximations for p_i in FB
Galois group almost certainly not weakly super solvableGaloisConj: Bound %Z
%d%% tschirnfilename of Term::Gnuplot does not end in `.pm': `%s'zero discriminant in Qfbnot a t_QFI in nucomplllint[1], kmax = %ldcount_max = %ld
lllfp[1], kmax = %ldzero polynomial forbiddensorting...
large exponent in Mod(a,N)^n: reduce n mod phi(N)bestappr0sfcont2Can't install Perl function with prototype `%s'Math::Pari::interface3gequalggegmulGGGGDGDGDempty vector in vecmaxBerlekamp_ker%ld apprpadicFqX_split_Trager: choosing k = %ld
no such error number: %ld %s: %s
,errpilegdiverGGD0,L,Strexpand({str}*): concatenates its (string) argument into a single string, performing tilde expansionalgdepbernrealGGD0,L,pcomponentellap(e,p,{flag=0}): computes a_p for the elliptic curve e using Shanks-Mestre's method. flag is optional and can be set to 0 (default) or 1 (use Jacobi symbols)ellsearch(N): if N is an integer, it is taken as a conductor else if N is a string, it can be a curve name ("11a1"), a isogeny class ("11a") or a conductor ("11"). Return all curves in the elldata database that match the  property.fordivgaloissubfieldsgetheapgetheap(): 2-component vector giving the current number of objects in the heap and the space they occupyidealdiv(nf,x,y,{flag=0}): quotient x/y of two ideals x and y in HNF in the number field nf. If (optional) flag is non-null, the quotient is supposed to be an integral ideal (slightly faster)idealfactoridealhnfidealstar(nf,I,{flag=1}): gives the structure of (Z_K/I)^*. flag is optional, and can be 0: simply gives the structure as a 3-component vector v such that v[1] is the order (i.e. eulerphi(I)), v[2] is a vector of cyclic components, and v[3] is a vector giving the corresponding generators. If flag=1 (default), gives idealstarinit, i.e. a 6-component vector [I,v,fa,f2,U,V] where v is as above without the generators, fa is the prime ideal factorisation of I and f2, U and V are technical but essential to work in (Z_K/I)^*. Finally if flag=2, same as with flag=1 except that the generators are also givenintnumrombintnumsteplngammaLDGnext({n=1}): interrupt execution of current instruction sequence, and start another iteration from the n-th innermost enclosing loopspoltchebiqfbnucomp(x,y,l): composite of primitive positive definite quadratic forms x and y using nucomp and nudupl, where l=[|D/4|^(1/4)] is precomputedqfbnupowreal(x): real part of xrnfidealhnf(rnf,x): relative version of idealhnf, where rnf is a relative numberfieldsigma(x,{k=1}): sum of the k-th powers of the divisors of x. k is optional and if omitted is assumed to be equal to 1sumnumalt*%ld 
#### Looking for random relations
Matrix of wrong dimensions in galoissubcycloroots_to_polacos(x)=inverse cosine of xbezout(x,y)=gives a 3-dimensional row vector [u,v,d] such that d=gcd(x,y) and u*x+v*y=dGD0.3,G,D0.3,G,D5,G,D1,G,D4,L,D3,L,pchar1(x,y)=det(y*I-x)=characteristic polynomial of the matrix x using Lagrange interpolationchinese(x,y)=x,y being integers modulo mx and my,finds z such that z is congruent to x mod mx and y mod mychptellcompocompositumcomprealrawdiscrayabscond(bnr,subgroup)=absolute [N,R1,discf] of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroup. Result is zero if fmodule is not the conductordiscrayabslist(bnf,listes)=if listes is a 2-component vector as output by ideallistunit or similar, gives list of corresponding discrayabscondgaussmodulogetrand()=current value of random number seedincgam3(s,x)=complementary incomplete gamma functionisprincipalraygenkaramul(x,y,k)=THIS FUNCTION HAS BEEN SUPPRESSEDlabelmin(x,y)=minimum of x and ynfreduce(nf,a,id)=gives r such that a-r is the ideal id and r is smallGGGD&V=GID1,L,pregula(x)=regulator of the real quadratic field of discriminant xrnfbasis(bnf,order)=given an order as output by rnfpseudobasis or rnfsteinitz, gives either a basis of the order if it is free, or an n+1-element generating setsmallfactsubcyclo(p,d)=finds an equation for the d-th degree subfield of Q(zeta_p), where p must be a prime powervec(x)=transforms the object x into a vector. Used mainly if x is a polynomial or a power seriesvecmin(x)=minimum of the elements of the vector/matrix xzidealloginteger too largelogfiledefault: incorrect value for %s [0:off / 1:on]FFInit: using subcyclo(%ld, %ld)
infinite group in forsubgroup  countsub = %ld
not a polynomial in polhenselliftsubresextinfinities of different type in intnuminitgengaddmatnegative size in fill_scalcolbreak not allowedcan't derive thisunknown function '%s', expected '=' instead of
rel = a(n)---- (type RETURN to continue) ----O(}{st(%c,expo=%ld):Mod(write failedincorrect precision in transcgpow: 0 to a non positive exponentnegative power in element_powid_mod_pnot a matrix in matbasistoalga log appears in intformaltwistpartialzeta (1), j = %ld/%ldbad argument for modular functionnot a pseudo-matrix in rnfdetk =   entering Dedekind Basis with parameters p=%Z
  leaving Decomp: f1 = %Z
f2 = %Z
e = %Z
de= %Z
not a pseudo-matrix in %suseless flag in rnfisnorm: the extension is GaloisplothsizesLGD0,L,not a suitable vector of elements in idealchinesenot a factorisation in decodemoduleWarning:accuracy problems          squfof [caller of] (n or 3n is a square)	But it nevertheless wasn't a %ld%s power.
	checking modulo %ld
factoring 0 in ifac_startIFAC: new partial factorization structure (%ld slots)
IFAC: main loop: another factor was divisible by
	%Z
IFAC: cracking composite
	%Z

MPQS: split N whilst combining, time = %ld ms
MPQS: Gauss elimination will require more than
	128MBy of memoryMPQS (relation is a nonsquare)MPQS panickingchanging f(x): p divides disc(g)
candidate = %Z
non-monic polynomial in galoisconj4galoisconj _may_ hang up for this polynomialhess, m = %ldEntering hnffinal:
partitions( %ld ) is meaninglessTerminal does not define put_textCan't load Gnuplot drawing engine from '%s': %sThis runtime link with gnuplot-shim does not implement midlevel start/end functionszero discriminant in %scornacchianegative bound in zncoppersmithexpected shvector bitsize: %ld
inconsistent primes in plindepnegative polynomial degree in algdeptime for ct = %ld : %ld
associationnegative integer in sqrtintroots must be bracketed in solvenon-monic polynomial in thuex2 -> %Z
* Checking for small solutions
Something very wrong  in %s, %iMath::Pari::pari2nvMath::Pari::interface1Math::Pari::interface16arg0, arg00, arg1=0, arg2=0, arg3=0gdivround_cosname, valnewsize = 0Berlekamp matrix   %3ld factor of degree %3ld
too many iterations in rootsspec_FqXQ_powduplicate indeterminates in reordertalkerO(a^b): p-adic or power series zero with precision given by bStrtexacos(x): inverse cosine of xcotandirdivellwpexpGLDnforvec(x=v,seq,{flag=0}): v being a vector of two-component vectors of length n, the sequence is evaluated with x[i] going from v[i][1] to v[i][2] for i=n,..,1 if flag is zero or omitted. If flag = 1 (resp. flag = 2), restrict to increasing (resp. strictly increasing) sequencesgaloisexport(gal,{flag}): gal being a galois field as output by galoisinit, output a string representing the underlying permutation group in GAP notation (default) or Magma notation (flag = 1)GGD0,L,Dngaloisidentify(gal): gal being a galois field as output by galoisinit, output the isomorphism class of the underlying abstract group as a two-components vector [o,i], where o is the group order, and i is the group index in the GAP4 small group libraryidealhnf(nf,a,{b}): hermite normal form of the ideal a in the number field nf, whatever form a may have. If called as idealhnf(nf,a,b), the ideal is given as aZ_K+bZ_K in the number field K defined by nfidealmulidealredidealtwoeltV=GGEDGpintlaplaceinv(X=sig,x,s,{tab}): numerical integration on the line real(z) = sig of s(z)exp(xz)dz/(2*I*Pi), i.e. inverse Laplace transform of s at x. tab is as in intnum.intnum(X=a,b,s,{tab}): numerical integration of s from a to b with  respect to X. a (and similarly b) is coded as follows. It can be a scalar: f is assumed to be C^infty at a. It can be a two component vector [a[1],a[2]], where a[1] is the scalar, and a[2] is the singularity exponent (in ]-1,0]), logs being neglected. It can be a one component vector [1] or [-1] meaning +infty or -infty, slowly decreasing functions. It can be a two component vector [[1], z] or [[-1], z], where [1] or [-1] indicates +infty or -infty and z is coded as follows. If z is zero, slowly decreasing. If z is real positive, exponentially decreasing, of the type exp(-zX). If z<-1, very slowly decreasing like X^(-z). If z is complex nonreal, real part is ignored and if z = r+I*s then if s>0, cosine oscillation exactly cos(sX), while if s<0, sine oscillation exactly sin(sX). If f is exponentially decreasing times oscillating function, you have a choice, but it is in general better to choose the oscillating part. Finally tab is either 0 (let the program choose  the integration step), a positive integer m (choose integration step 1/2^m), or a table tab precomputed with intnuminit (depending on the type of interval: compact, semi-compact or R, very slow, slow, exponential, or cosine or sine-oscillating decrease).V=GGED0,L,pmatmultodiagonalmatsupplement(x): supplement the columns of the matrix x to an invertible matrixnfkermodprnfrootspolgaloispolrootspadicLLprinttexqfjacobiqflll(x,{flag=0}): LLL reduction of the vectors forming the matrix x (gives the unimodular transformation matrix). The columns of x must be linearly independent, unless specified otherwise below. flag is optional,  and can be 0: default, 1: assumes x is integral, columns may be dependent, 2: assumes x is integral, returns a partially reduced basis, 4: assumes x is  integral, returns [K,I] where K is the integer kernel of x and I the LLL reduced image, 5: same as 4 but x may have polynomial coefficients, 8: same as 0 but x may have polynomial coefficientsremoveprimes({x=[]}): remove primes in the vector x (with at most 100 components) from the prime table. x can also be a single integer. List the current extra primes if x is omittedrnfidealtwoeltrnfidealtwoelt(rnf,x): relative version of idealtwoelt, where rnf is a relative numberfieldrnflllgram(nf,pol,order): given a pol with coefficients in nf and an order as output by rnfpseudobasis or similar, gives [[neworder],U], where neworder is a reduced order and U is the unimodular transformation matrixwritetexfloating point exception: bug in PARI or calling programprecision too low for generators, not given
#### Computing class group generators
cleanarchbernvec(x)=Vector of rational Bernoulli numbers B_0, B_2,... up to B_(2x)binbuchreal(D,...)=compute the structure of the class group and the regulator of the real quadratic field of discriminant D>0 in the wide sense. See manual for the other parameters (which can be omitted)coeffcoredisc(n)=discriminant of the quadratic field Q(sqrt(n))detdiscrayrel(bnr,subgroup)=relative [N,R1,rnfdiscf] of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroupdivreseigen(x)=eigenvectors of the matrix x given as columns of a matrixextract(x,y)=extraction of the components of the vector x according to the vector or mask y, from left to right (1, 2, 4, 8, ...for the first, second, third, fourth,...component)fact(x)=factorial of x (x C-integer), the result being given as a real numbergaloisconjforcehclassnohvector(n,X,expr)=row vector with n components of expression expr, the variable X ranging from 1 to nincgam1indexrankisincl(x,y)=tests whether the number field defined by the polynomial x is isomorphic to a subfield of the one defined by y; 0 if not, otherwise all the isomorphismsorderellpascal(n)=pascal triangle of order n (n C-integer)perf(a)=rank of matrix of xx~ for x minimal vectors of a gram matrix apolylogp(m,x)=P_m-modified m-th polylog of xpsi(x)=psi-function at xredrealnod(x,sq)=reduction of the binary quadratic form x with D>0 without distance function where sq=[sqrt D]rootsold(x)=roots of the polynomial x using a modified Newton's methodsigmak(k,x)=sum of the k-th powers of the divisors of x (k C-integer)smallbasissubst(x,y,z)=in expression x, replace the variable y by the expression zreducebeta(warn when using obsolete functions)(TeX output)psfileIFAC: Stop: Primary factor: %Z
Rg_to_Fpbound 2^%ld. Goal 2^%ld(lifted) subgp of prime to %Z part:
subgroup: index bound must be positiveBeauzamy bound: %Z
negative size in fill_scalmatusing obsolete function %sray torsion unitsnew precision: %ld
D(6) = S(3)[x]2galois (bug2)not an integer type in dirzetaklog(n)[;]
I/O: removed file %s
^%ldunknown user nth-root does not exist in gsqrtn[apell1] sortingincorrect archimedean component in Idealstarnot an element in zideallogincorrect permtuation to inversevaluation not equal to 1 in serreversenegative index in polylog--> %Z : reduced absolute generator: %Z
  dedek: gcd has degree %ld
initial parameters p=%Z,
  f=%Z
bug in Decomp (not a factor), is p = %Z a prime?%s

couldn't open dynamic library '%s'plotrline(w,dx,dy): if the cursor is at (x1,y1), draw a line from (x1,y1) to (x1+dx,y1+dy) (and move the cursor) in the rectwindow wvLsD0,L,ideal_two_eltnon-integral exponent in idealpownot a matrix in %snot a matrix of maximal rank in nfsmithnot a vector in idealred  Testing primes <= B (= %lu)

wrong type in too_bigThis is not a valid color	(extracted precomputed helix / baby step entries)
SQUFOF: blacklisting a = %ld on first cycle
OddPwrs: examining %Z
ifac_decompifac_omegaIFAC: found %Z =
	%Z ^%ld
factoringfactor has NULL exponent in ifac_find
MPQS: found factor = %ld whilst creating factor base
MPQS: bound for `large primes' = %ld
MPQS: Net yield: %4.3g full relations per 100 candidates
MPQS: found %ld factors =
MPQS: kN = %Z
MPQS: precomputing auxiliary primes up to %ld
MPQS: new bit pattern for primes for A: 0x%lX
GaloisConj:val1=%ld val2=%ld
GaloisConj:Testing A4 first
testpermutation(%Z)Not a group in group_identQuickNormL1
hnfall, final phase: Leaving hnffinal
%s/galdata/NAM%ldElliptic curves files not available for conductor %ld
[missing %s]panic: more than %d tokens for optionsNo terminal specifiedCannot reset output routines to copy term list to a variable.pm
redimagsl2not a definite form in minim0isanypower: now k=%ld, x=%Z
1/0 exponent in Fp_sqrtnincorrect bound type in bestapprMath::Pari::EXISTSMath::Pari::pari2pvMath::Pari::interface_flexible_intMath::Pari::interface13Math::Pari::dumpHeapMath::Pari::PARI_DEBUG_setToo many args for a flexible-interface functionCannot process default argument %.*s of type %.1s for prototype '%s'gsubUnsupported interface %d for "direct-link" Pari function %s_flexible_longCannot load a Pari macro `%s': macros are unsupported; VALENCE=%#04x, code=<%s>, isFunction=%d, EpVAR=%dheap had %ld bytes (%ld items)
digits=0too many iterations in hqr%s: Time ListStrarg(x): argument of x,such that -pi<arg(x)<=piatanh(x): inverse hyperbolic tangent of xLpbigomega(x): number of prime divisors of x, counted with multiplicitybnfunitcharpoly(A,{v=x},{flag=0}): det(v*Id-A)=characteristic polynomial of the matrix or polmod A. flag is optional and may be set to 1 (use Lagrange interpolation) or 2 (use Hessenberg form), 0 being the defaultconjvecellchangecurve(x,y): change data on elliptic curve according to y=[u,r,s,t]GD0,L,pellj(x): elliptic j invariant of xellsub(e,z1,z2): difference of the points z1 and z2 on elliptic curve eevalfloor(x): floor of x = largest integer<=xforellintfouriercosintfourierexpintlaplaceinvnfalgtobasis(nf,x): transforms the algebraic number x into a column vector on the integral basis nf.zknfbasis(x,{flag=0},{p}): integral basis of the field Q[a], where a is a root of the polynomial x, using the round 4 algorithm. Second and third args are optional. Binary digits of flag mean 1: assume that no square of a prime>primelimit divides the discriminant of x, 2: use round 2 algorithm instead. If present, p provides the matrix of a partial factorization of the discriminant of x, useful if one wants only an order maximal at certain primes onlynfrootsof1(nf): number of roots of unity and primitive root of unity in the number field nfpermtonumLDnpolisirreduciblepollead(x,{v}): leading coefficient of polynomial or series x, or x itself if x is a scalar. Error otherwise. With respect to the main variable of x if v is omitted, with respect to the variable v otherwisepollegendre(n,{v=x}): legendre polynomial of degree n (n C-integer), in variable vqfbsolvequadgen(x): standard generator of quadratic order of discriminant xrnfidealdownrnfidealnormrel(rnf,x): relative norm of the ideal xrnfisnormsizebytesqrsuminftanvectorsmall(n,{X},{expr=0}): VECSMALL with n components of expression expr (X ranges from 1 to n) which must be small integers. By default, fill with 0svectorv(n,{X},{expr=0}): column vector with n components of expression expr (X ranges from 1 to n). By default, fill with 0sznorder(x,{o}): order of the integermod x in (Z/nZ)*. Optional o is assumed to be a multiple of the order.SPLIT: increasing factor base [%ld]
makematalclassgroup generators(%ld)for this relation
#### Computing regulator multiple
Be honest for %ld primes from %ld to %ld
SubCyclo: new conductor:%ld
boundcfbuchcertify(bnf)=certify the correctness (i.e. remove the GRH) of the bnf data output by buchinit or buchinitfucore2(n)=(long)gen_2-component row vector [d,f], where d is the unique squarefree integer dividing n such that n/d=f^2 is a squarecos(x)=cosine of xdeplin(x)=finds a linear dependence between the columns of the matrix xextractfibo(x)=fibonacci number of index x (x C-integer)forprime(X=a,b,seq)=the sequence is evaluated, X running over the primes between a and bgaloisapply(nf,aut,x)=Apply the Galois automorphism sigma (polynomial or polymod) to the object x (element or ideal) in the number field nfgaloisconjgetheap()=2-component vector giving the current number of objects in the heap and the space they occupyhermitehilb(x,y,p)=Hilbert symbol at p of x,y (integers or fractions)idealapprfact(nf,x)=x being a prime ideal factorization with possibly zero or negative exponents, gives an element b such that v_p(b)=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other pidealmulred(nf,x,y)=reduced product of the two ideals x and y in the number field nfidealpow(nf,x,n)=n-th power of the ideal x in HNF in the number field nfintgenjellnfdiveuc(nf,a,b)=gives algebraic integer q such that a-bq is smallnupow(x,n)=n-th power of primitive positive definite quadratic form x using nucomp and nuduplpolredabs(x)=a smallest generating polynomial of the number field for the T2 norm on the roots, with smallest index for the minimal T2 normprime(n)=returns the n-th prime (n C-integer)principalidele(nf,x)=returns the principal idele generated by the algebraic number x in the number field nfrank(x)=rank of the matrix xrayclassnoresultant(x,y)=resultant of the polynomials x and y with exact entriesresultant2(x,y)=resultant of the polynomials x and yrootmodsmithpolsolve(X=a,b,expr)=real root of expression expr (X between a and b), where expr(a)*expr(b)<=0tschirnhausuntil(a,seq)=evaluate the expression sequence seq until a is nonzeroveclexsort(x): sort the elements of the vector x in ascending lexicographic orderkummer for composite relative degreepolrel(beta) = %Z
debug   [logfile was "%s"]
(bits 0x2/0x4 control output of \left/\PARIbreak)factor_add_primesprime %Z is not in primetablenon-invertible polynomial in FpXQX_gcdMignotte bound: %Z
sumnuminit0empty matrix in deplina6indexr1not an integerbreak not allowed in test expressionglobal variable: %ld^%ld cglob = %ld. Bach constant <= 0 in buchquadget_lambda
Time %s rel [#rel/#test = %ld/%ld]: %ld
ray unitschecknfi0 = %ld
t_INTt_QUADmalformed binary file (no name)gen_0
chars:TMPDIRpowell for non integral, non CM, exponentselement_sqriscomplexvariable must have higher priority in gtopolynegative or zero constant in veceint1allbaseunknown identifier valence, please reportgtodblList for up to %ld roundsSQUFOF: second cycle exhausted after %ld iterations,
	dropping it
 5th%sfactoring 0 in ifac_decompMPQS: striving for %ld relations
MPQS: smallest prime used for sieving FB[%ld] = %ld
MPQS: factoring this number will take %s hours:
N = %Zcannot rename file %s to %s
* Look for subfields of degree %ld

Y = %Z
GaloisConj:%d hop sur %Z iterations
galoisindex for groups of order >127matrix of non-maximal rank in matrixqzmit = %Z
B = %Z
C = %Z
ODD	# rational integer roots = %ld:minim0not an element of (Z/nZ)* in ordernon integral index in suminf  c10 = %Z
Math::Pari::FETCHMath::Pari::loadPariMath::Pari::setprecisionMath::Pari::longwordAssignment of a columns into a matrix of incompatible height_gcmpGDVDIV=GEpV=GGEDrvx, nabs is not meromorphic at 0error in rootsold(): using roots2()lx<ly in Flx_addmul_inplacefactmod: %lu is not prime* Finding eigenvalues
changevaruncaught error: %ldasinh(x): inverse hyperbolic sine of xbnfdecodemodule(nf,fa): given a coded module fa as in bnrdisclist, gives the true moduleellgeneratorsellheightmatrixellordinateellzeta(om,z): om=[om1,om2], value of the Weierstrass zeta function of the lattice generated by om at zfactorial(x): factorial of x (x C-integer), the result being given as a real numberfactorint(x,{flag=0}): factor the integer x. flag is optional, whose binary digits mean 1: avoid MPQS, 2: avoid first-stage ECM (may fall back on it later), 4: avoid Pollard-Brent Rho and Shanks SQUFOF, 8: skip final ECM (huge composites will be declared prime)frac(x): fractional part of x = x-floor(x)lGGDGvSlistkill(list): kills listmatdetint(x): some multiple of the determinant of the lattice generated by the columns of x (0 if not of maximal rank). Useful with mathnfmodmatsolvematsolvemod(M,D,B,{flag=0}): one solution of system of congruences MX=B mod D (M matrix, B and D column vectors). If (optional) flag is non-null return all solutionsnfeltmul(nf,a,b): element a. b in nfpolredordprimeprintp(a): outputs a (in beautified format) ending with newlineprod(X=a,b,expr,{x=1}): x times the product (X runs from a to b) of expressionsolvethue(tnf,a,{sol}): solve the equation P(x,y)=a, where tnf was created with thueinit(P), and sol, if present, contains the solutions of Norm(x)=a modulo units in the number field defined by P. If tnf was computed without assuming GRH (flag 1 in thueinit), the result is unconditionaltype(x): return the type of the GEN x.rfix (conversion to t_REAL)not an algebraic number in isunit# ideals tried = %ld
weighted G matrices  small norms gave %ld relations.
vector in new basis = %Z
SubCyclo: %ld not found
Please do not try to break PARI with ridiculous counterfeit data. Thanks!basis2buchimagcenterlift(x)=centered lift of x. Same as lift except for integermodscharGVIGLLforvec(x=v,seq)=v being a vector of two-component vectors of length n, the sequence is evaluated with x[i] going from v[i][1] to v[i][2] for i=n,..,1hell(e,x)=canonical height of point x on elliptic curve E defined by the vector e computed using theta-functionshermite(x)=(upper triangular) Hermite normal form of x, basis for the lattice formed by the columns of x, using a naive algorithmidealnorm(nf,x)=norm of the ideal x in the number field nfidealtwoelt2image2(x)=basis of the image of the matrix xincgam3lllintpartialminim(x,bound,maxnum)=number of vectors of square norm <= bound, maximum norm and list of vectors for the integral and definite quadratic form x; minimal non-zero vectors if bound=0modpolylogpprod(x,X=a,b,expr)=x times the product (X runs from a to b) of expressionquadpoly(x)=quadratic polynomial corresponding to the discriminant xreverse(x)=reversion of the power series xsumalt2sumpos2trans(x)=x~=transpose of xveczideallog(nf,x,bid)=if bid is a big ideal as given by zidealstarinit or zidealstarinitgen , gives the vector of exponents on the generators bid[2][3] (even if these generators have not been computed) (%ld digits displayed)   format = %c%ld.%ld
nodarkbg0(off)PARIpromptENDoutputbroken prettyprinter: '%s'Rg_to_FpXQpows [P,Q]non positive degree in ffinit    lambda'= need positive degree in gdeflatelifting to prec %ld
### K = %d, %Z combinations
variable number too bigfutuseqsymbol already in use
*** Bach constant: %f
class number = %ldargument must be positive in polcycloprecision<=0 in gprecconductor too large in ArtinNumbercheckbnfplease apply bnrinit(,,1) and not bnrinit(,)incompatible modulus in %s:
  mod = %Z,
  nf  = %ZC(5) = 5nf_ADDZK flag when nf_ALL set (polredabs)need %Z coefficients in initzeta: computation impossiblenot a zeta number field in zetakallget_polgp_readvec_stream: found %ld entries
t_INTMODinvalid range in print_functions_hash.:~:~/gpYou never gave me anything to read!nd%ld%s component = qfr( ( +odd exponent in p-adic sqrtsecond arg must be integer in gsqrtnnot an element of (Z/pZ)* in znlogIdealstar needs an integral non-zero ideal: %Zinvalid data in qf_base_changenon existent component in truecoeffgaloissubgroup: not a WSS groupnf_factor_boundnf_to_ffnot the same variable in compositum new order:
%Z
%Z
plotcursor(w): current position of cursor in rectwindow wploth(X=a,b,expr,{flags=0},{n=0}): plot of expression expr, X goes from a to b in high resolution. Both flags and n are optional. Binary digits of flags mean: 1=Parametric, 2=Recursive, 4=no_Rescale, 8=no_X_axis, 16=no_Y_axis, 32=no_Frame, 64=no_Lines (do not join points), 128=Points_too (plot both lines and points), 256=Splines (use cubic splines), 512=no_X_ticks, 1024= no_Y_ticks, 2048=Same_ticks (plot all ticks with the same length). n specifies number of reference points on the graph (0=use default value). Returns a vector for the bounding boxplothrawplotlinetypevLGGGGlsBuchrayMinkowski bound is too large[ %ld, %ld, %ld ]: %Z
  *** testing p = %lu
not a polynomialinfinite precision	(giant step at p = %lu)
Rho: using X^2%+1ld for up to %ld rounds of 32 iterations
IFAC: cofactor = %Z
prime equals composite in ifac_sort_one    {%ld @ %s : %s}
  * {%ld @ %s : %s}
MPQS: X^2 - Y^2 != 0 mod N
MPQS: wrong relation found after Gaussfield not Galois or Galois group not weakly super solvableGaloisAnalysis:p=%ld l=%ld group=%ld deg=%ld ord=%ld
GaloisConj:Sortie Init Test
A4GaloisConj:O=%Z 
hnflll, k = %ld / %ldhnfall
i = %ld: name "%s" for terminal too longTerminal does not define resetperl -MConfig -wle %cuse Term::Gnuplot;print $INC{qq(Term/Gnuplot.pm)};print $Config{dlext}%cnot a prime in Fp_sqrtstep equal to zero in forstepratlift failed to catch d1 == 0
Import of Perl function with too many argumentsMath::Pari::interface34Math::Pari::interface73Math::Pari::pari2boolsvCalling Perl via PARI with an unknown interface: avoiding looparg1, arg2=0, arg3=gen_0not an integer modulus in cvtoppopping %s (bloc no %ld)
 in %s; new prec = %ld
siginterModStrexpandabsbesselh1GDGpbnfisprincipaldenominator(x): denominator of x (or lowest common denominator in case of an array)ellisoncurve(e,x): true(1) if x is on elliptic curve e, false(0) if notellrootno(e,{p=1}): root number for the L-function of the elliptic curve e. p can be 1 (default), global root number, or a prime p (including 0) for the local root number at pellsigmaellzetaGDGD0,L,Dnidealaddidealfactor(nf,x): factorization of the ideal x given in HNF into prime ideals in the number field nfmatisdiagonalnfdiscnorm(x): norm of xnumdiv(x): number of divisors of xnumerator(x): numerator of xrnfbasisrnfdisc(nf,pol): given a pol with coefficients in nf, gives a 2-component vector [D,d], where D is the relative ideal discriminant, and d is the relative discriminant in nf^*/nf*^2rnfpseudobasissetunionshiftmul(x,n): multiply x by 2^n (n>=0 or n<0)GD1,L,substpolsubstpol(x,y,z): in expression x, replace the polynomial y by the expression z, using remainder decomposition of x.sumpostruncatevariable(x): main variable of object x. Gives p for p-adic x, error for scalarsgchnon-positive integer in glngammaBach constant <= 0 in buchadding vector = %Z
bestappr(x,k)=gives the best approximation to the real x with denominator less or equal to kboundfact(x,lim)=partial factorization of the integer x (using primes up to lim)buchinitfubuchrayinitgen(bnf,ideal)=given a big number field as output by buchinitfu (only) and  an ideal or a 2-component row vector formed by an ideal and a list of R1 zeros or ones representing a module, initializes data for computing in the ray class group  corresponding to this module. In particular, the fifth component is the ray class group structureceil(x)=ceiling of x=smallest integer>=xchangevar(x,y)=change variables of x according to the vector yconvoldiscrayrelfactoredpolredidealapprfactidealhermitejacobi(x)=eigenvalues and orthogonal matrix of eigenvectors of the real symmetric matrix xminideal(nf,ix,vdir)=minimum of the ideal ix in the direction vdir in the number field nfnumerpf(x,p)=returns the prime form whose first coefficient is p, of discriminant xprimedec(nf,p)=prime ideal decomposition of the prime number p in the number field nf as a vector of 5 component vectors [p,a,e,f,b] representing the prime ideals pZ_K+a.Z_K, e,f as usual, a as vector of components on the  integral basis, b Lenstra's constantresultant2rhorealnod(x,sq)=single reduction step of the binary quadratic form x with D>0 without distance function where sq=[sqrt D]simplefactmodsizesmallfact(x)=partial factorization of the integer x (using only the stored primes)srgcdStep 2
Step 13
[%ld,%ld,%ld]readline   prettyprinter = "%s"
formatfflgenPolynomials not irreducible in FpX_ffintersectKnapsackmissing nf in factorbackeltqrom2: iteration %ld: %Z
no such component in vecextractEntering gauss with inexact=%ld
gauss_pivot. k=%ld, n=%lda stringified flag does not start with an ida4wunknown member function

KC = %ld, need %ld relations
qrf5_rho_powincorrect bigray fieldC(4) = 4D_6(6) = [3]2 Used :                         %ld  long words  (%ld K)
 Available :                    %ld  long words  (%ld K)
]
}
}%s is a GP binary file. Please use writebin%.8s%snot a p-adic argument in teichmullergcosnot a rational curve in ellintegralmodelellinit data not accurate enough. Increase precisionnot the same number field in algtobasisffshankstreating pr^%ld, pr = %Z
evaluation of a power series>@rnf function[secure mode]: about to install '%s'. OK ? (^C if not)
vLD0,G,D0,G,D0,L,zero ideal in idealfactorquotient not integral in idealdivexactnot a module in %scan't deflate%ld %ld l
(%s	Pollard-Brent failed.
SQUFOF: giving up, time = %ld ms
 7th	stored (largest) factor no. %ld...
 == {%s}
	index i = %ld
MPQS: got two factors, looking for more...
unknown2 * M = %Z
please apply galoisinit firstGaloisConj:denominator:%Z
GaloisConj: G[%d]=%Z of relative order %d
A4GaloisConj:orb=%Z 
Galoisconj:Subgroups list:%Z
Classification of transitive groups of order > 30 is not knownsmithallaccuracy lost in matfrobeniusmatgen = %Z
Shanks distance must be a t_REAL in qfrreducible form in qfr_initk =dimension 0 in fincke_pohstNew bound: %Zroots (conjugates)negative number of columns in matrixnon integral index in sumaltAll solutions are <= %Z
PARI: %sMath::Pari::_to_int_flexible_voidGGGD0,L,pV=GGEGGDVDVD# gaffect (gen_2)gaffect (gi)normalizepollbot>ltop in gerepileDGMat({x=[]}): transforms any GEN x into a matrix. Empty matrix if x is omittedbezoutresbitnegbittestcontfracGDGDGvV=GDGIidealdivGGGD0,L,intcirc(X=a,R,s,{tab}): numerical integration of s on the circle  |z-a|=R, divided by 2*I*Pi. tab is as in intnum.lcm(x,{y}): least common multiple of x and y, i.e. x*y / gcd(x,y)GGLlistsort(list,{flag=0}): sort list in place. If flag is non-zero, suppress all but one occurence of each element in listlogmatkerintnextprime(x): smallest pseudoprime >= xnfeltdivrem(nf,a,b): gives [q,r] such that r=a-bq is smallnfroots({nf},pol): roots of polynomial pol belonging to nf (Q if omitted) without multiplicitypadicappr(x,a): p-adic roots of the polynomial x congruent to a mod pprimes(n): returns the vector of the first n primes (n C-integer)prodeulerrnfbasistoalg(rnf,x): relative version of nfbasistoalg, where rnf is a relative numberfieldsetissetweberGGGDGPHASE 1: check primes to Zimmert bound = %lu

*** p = %lu
fundamental units too largebuchgen(P,...)=compute the structure of the class group and the regulator for the number field defined by the polynomial P. See manual for the other parameters (which can be omitted)V=GGIDGdiscf(x)=discriminant of the number field defined by the polynomial x using round 4euler=euler()=euler's constant with current precisiongamh(x)=gamma of x+1/2 (x integer)gaussgaussmodulo(M,D,Y)=(long)gen_1 solution of system of congruences MX=Y mod Didealappr(nf,x)=x being a fractional ideal, gives an element b such that v_p(b)=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other pidealpowredinitalginitzetainverseimage(x,y)=an element of the inverse image of the vector y by the matrix x if one exists, the empty vector otherwiseisdiagonal(x)=true(1) if x is a diagonal matrix, false(0) otherwiseisqrt(x)=integer square root of x (x integer)issqfree(x)=true(1) if x is squarefree, false(0) if notjbesselhlllrat(x)=lll reduction of the vectors forming the matrix x, computations done with rational numbers (gives the unimodular transformation matrix)matnfmul(nf,a,b)=element a.b in nfLGpprimes(n)=returns the vector of the first n primes (n C-integer)reorder(x)=reorder the variables for output according to the vector xresultantrnfsteinitz(nf,order)=given an order as output by rnfpseudobasis, gives [A,I,..] where (A,I) is a pseudo basis where all the ideals except perhaps the last are trivialsetunion(x,y)=union of the sets x and ysmithcleansort(x)=sort in ascending order of the vector xtaniyama(e)=modular parametrization of elliptic curve evecindexsort(x): indirect sorting of the vector xisvirtualunitsignificant terms(bits 0x2/0x4 control matched-insert/arg-complete)lineshistsize   help = "%s"
IFAC: Stop: remaining %Z
resultant mod %ld (bound 2^%ld, stable = %d)QXQ_inv %ld factor for general polynomialsspecial_pivot output:
%Z
Fourier transform of oscillating functionsincorrect beginning value in sumnumincorrect length in sumhnfdividemissing eigenspace. Compute the matrix to higher accuracy, then restart eigen at the current precisionid too long in a stringified flaga1a2[+++]unknown parser codeuser function %s: variable %Z declared twiceform_to_idealindex too large in vecsortcpl = 2^%ld
L(6):2 = PGL(2,5) = S_5(6)Found %ld minimal polynomials.
too many terms in dirzetaks = 1 is a pole (gzetakall)[;]matrix(0,%ld,j,k,0)\*gpow: underflow or overflowp-adic argument out of range in gexpcan't evaluate log(ellsigma) at lattice pointinversenonexistent component    pass no %ld
plotpointtype(w,type): change the type of following points in rectwindow w. type -1 corresponds to a dot, larger values may correspond to something else. w=-1 changes highlevel plottingplotrboxnot a prime ideal factorization in idealchinesewhat's going on ?rectdraw	(got [2]Q...[10]Q)
, oravoiding nonexistent factors in ifac_whoiswhoMPQS: %4.1f%% of the polynomials yielded no candidates
several FB[%ld]=%ld%s%s @ %s :%s
[1]: mpqs_solve_linear_systemTR_POL(-1), i = %ld/%ldgaloisanalysis()Entering hnfspec
non integral matrix in smithall
    Output of isin_%ld_G_H(%ld,%ld): %ldUnsupported function map3d_xy calledTerminal does not define pointqfr_unit_by_discnpass = %ld, red. last time = %ld, log_2(det) ~ %ld

delta = %d, t = %d, cond = %lf
Math::Pari::FETCHSIZEMath::Pari::set_gnutermMath::Pari::setprimelimitMath::Pari::added_sectionscv, name, numargs = 1, help = NULLExpected VOID return type, got code '%s'Unexpected syntax of default argument '%s' in prototype '%s'arg1arg1, arg2arg1, arg2=0, arg3=0_sinGGD0,Lfactorpadic2 for non-monic polynomialentering recover(), loc = %ld
Iacoshaliasbesseljh(n,x): J-bessel function of index n+1/2 and argument x, where n is a non-negative integerlGbnfclgp(P,{tech=[]}): compute the class group of the number field defined by the polynomial P. If P is a non-zero integer, it is interpreted as a quadratic discriminant. See manual for details about techbnfisunitdivisors(x): gives a vector formed by the divisors of x in increasing orderdivremellconvertnameellinitGGDGpellorder(e,p): order of the point p on the elliptic curve e over Q, 0 if non-torsionellsearchfactorcantorfactorcantor(x,p): factorization mod p of the polynomial x using Cantor-Zassenhausgaloisinitgaloissubcyclogammah(x): gamma of x+1/2 (x integer)idealprincipal(nf,x): returns the principal ideal generated by the algebraic number x in the number field nfintfuncinitintmellininvshortlistinsert(list,x,n): insert x at index n in list, shifting the remaining elements to the rightmathnfmodid(x,d): (upper triangular) Hermite normal form of x concatenated with d times the identity matrixmatid(n): identity matrix of order n (n C-integer)minmoebius(x): Moebius function of xnfhilbertlGGGDGnfhnfmod(nf,x,detx): if x=[A,I], and detx is a multiple of the ideal determinant of x, gives a pseudo-basis of the module sum A_jI_jqfminimrnfalgtobasisrnfdedekindrnfeltreltoabsrnfeltupround(x,{&e}): take the nearest integer to all the coefficients of x. If e is present, do not take into account loss of integer part precision, and set e = error estimate in bitssetintersectsetsearchsumnumtanhvecminvectorvgathproduct from 0 to N-1insufficient precision for fundamental unitsprecision too low for generators, e = %ldcompleting bnf (building matal)
 ***** check = %f
znstar_cosetsanell(e,n)=computes the first n Fourier coefficients of the L-function of the elliptic curve e (n<32768)buchgenbuchgenforcefu(P,...)=compute the structure of the class group, the regulator a primitive root of unity and a system of fundamental units for the number field defined by the polynomial P, and insist until the units are obtained. See manual for the other parameters (which can be omitted)cf2compositum2(pol1,pol2)=vector of all possible compositums of the number fields defined by the polynomials pol1 and pol2, with roots of pol1 and pol2 expressed on the compositum polynomialscyclofibogamhideallllred(nf,x,vdir)=LLL reduction of the ideal x in the number field nf along direction vdir, in HNFimag(x)=imaginary part of xker(x)=basis of the kernel of the matrix xkro(x,y)=kronecker symbol (x/y)lllgramintnfdiv(nf,a,b)=element a/b in nfnfpow(nf,a,k)=element a^k in nfpolredabs2GV=GGIrnfdiscfrnfequation2rnfhermitebasissmallpolred2(x)=partial reduction of the polynomial x (gives elements and minimal polynomials)torsell(e)=torsion subgroup of elliptic curve e: order, structure, generatorsvvectorwhile(a,seq)=while a is nonzero evaluate the expression sequence seq. Otherwise 0zidealstarinitgen(nf,I)=6-component vector [I,v,fa,f2,U,V] where v is as in zidealstar fa is the prime ideal factorisation of I and f2, U and V are technical but essential to work in (Z_K/I)^*secure? negative exponent in bitwise negationbitwise andbitwise xorpowers is only [] or [1] in FpX_FpXQV_compoS_2   bound: %Z^%ld
RgX_gcd_simplenfgcd: p=%d
MultiLift: bad argsnot an integer matrix in detintgauss_pivotb4elocal(); quadray needs a fundamental discriminantinitialnot a series in convolno data in polinterpolate2A_4(6) = [2^3]3 = 2 wr 3S_4(6c) = 1/2[2^3]S(3)F_36(6):2 = [S(3)^2]2 = S(3) wr 2galois (bug1)polredabs0 %ld objects on heap occupy %ld long words

I/O: new pariFILE %s (code %d) 
  I : 0.GPTMPDIRellpointtoz: %s square root
prime too large in apell2, use apellnot an integral model in akellpowell for nonintegral CM exponenttorsell (bug3)Pohlig-Hellman: DL mod %Z^%ld
gvarnot the same prime in padicprecnonexistent component in truecoeffinvalid vector in hqfevalGroup(original absolute generator: %Z
 treating %Z
plothsizes({flag=0}): returns array of 6 elements: terminal width and height, sizes for ticks in horizontal and vertical directions, width and height of characters. If flag=0, sizes of ticks and characters are in pixels, otherwise are fractions of the screen sizeplotkill(w): erase the rectwindow wplotmovepsplothgeneric conversion to finite fieldpol = %Z
Mahler bound for regulator: %Z
bnrdiscraynon invertible matrix in gaussrectpoints%.5gRho: restarting for remaining rounds...
 orifac_totientifac_sumdivkSQUFOF: found factor %ld from ambiguous form
	after %ld steps on the ambiguous cycle, time = %ld ms
Jacobi sums and tables computed
MPQS: sizing out of tune, FB too small or
	way too few primes in A :f = %Z
pol. found = %Z
GaloisConj:Back to Earth:%Z
not a rational matrix in matrixqzgalois files %s not compatible
RES %d K%ldalgdep0maximal number of vectors must be providedfinal sort & check...
FFTprimitive root mod 2^%ld does not existMath::Pari::interface21Math::Pari::interface2199Math::Pari::interface24Math::Pari::interface29_oldMath::Pari::interface84;$Math::Pari::lgefintExpected int return type, got code '%s'_sqrt_flexible_genvV=GID0,L,LV=GGIpD0,L,D0,L,%s %2ld: %s
forbidden divisor %Z in ggvalgaffect (gen_m1)polfnf: choosing k = %ld
  ***   Error in the PARI system. End of program.
Set({x=[]}): convert x into a set, i.e. a row vector with strictly increasing coefficients. Empty set if x is omittedaddprimesatan(x): inverse tangent of xbnrconductor(a1,{a2},{a3},{flag=0}): conductor f of the subfield of the ray class field given by a1,a2,a3 (see bnrdisc). flag is optional and can be 0: default, 1: returns [f, Cl_f, H], H subgroup of the ray class group modulo f defining the extension, 2: returns [f, bnr(f), H]bnrisconductorbnrstarkbreak({n=1}): interrupt execution of current instruction sequence, and exit from the n innermost enclosing loopscenterlift(x,{v}): centered lift of x. Same as lift except for integermodsellglobalred(e): e being an elliptic curve, returns [N,[u,r,s,t],c], where N is the conductor of e, [u,r,s,t] leads to the standard model for e, and c is the product of the local Tamagawa numbers c_petagaloisinit(pol,{den}): pol being a polynomial or a number field as output by nfinit defining a Galois extension of Q, compute the Galois group and all neccessary informations for computing fixed fields. den is optional and has the same meaning as in nfgaloisconj(,4)(see manual)galoissubgroupsidealintersect(nf,x,y): intersection of two ideals x and y in the number field defined by nfidealstarmatmuldiagonal(x,d): product of matrix x by diagonal matrix whose diagonal coefficients are those of the vector d, equivalent but faster than x*matdiagonal(d)numtopermLGpoldiscreducedrnfhnfbasis(bnf,order): given an order as output by rnfpseudobasis, gives either a true HNF basis of the order if it exists, zero otherwisernfisnorminit(pol,polrel,{flag=2}): let K be defined by a root of pol, L/K the extension defined by polrel. Compute technical data needed by rnfisnorm to solve norm equations Nx = a, for x in L, and a in K. If flag=0, do not care whether L/K is Galois or not; if flag = 1, assume L/K is Galois; if flag = 2, determine whether L/K is Galoisserconvol(x,y): convolution (or Hadamard product) of two power seriessqrtGGD&puntil(a,seq): evaluate the expression sequence seq until a is nonzerobus error: bug in PARI or calling programLIMC = %ld, LIMC2 = %ld
buchall (%s)*** Changing sub factor base
apell(e,p)=computes a_p for the elliptic curve e using Shanks-Mestre's methodarg(x)=argument of x,such that -pi<arg(x)<=pibytesizeclassno2(x)=class number of discriminant xdecodemoduledenom(x)=denominator of x (or lowest common denominator in case of an array)discrayabsdiscrayrelcond(bnr,subgroup)=relative [N,R1,rnfdiscf] of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroup. Result is zero if module is not the conductorfactorpadic2(x,p,r)=p-adic factorization of the polynomial x to precision r, using Buchmann-Lenstrafactpolhermite2hilbp(x,y)=Hilbert symbol of x,y (where x or y is integermod or p-adic)hyperu(a,b,x)=U-confluent hypergeometric functionideallistunitgen(bnf,bound)=2-component vector [L,U] where L is as ideallistzstar, and U is a vector of vector of zinternallogs of the units, with generatorsidealpowred(nf,x,n)=reduced n-th power of the ideal x in HNF in the number field nfif(a,seq1,seq2)=if a is nonzero, seq1 is evaluated, otherwise seq2isfundlegendre(n)=legendre polynomial of degree n (n C-integer)lllint(x)=lll reduction of the vectors forming the matrix x when the gram matrix is integral (gives the unimodular transformation matrix)matrixqz(x,p)=transforms the rational or integral mxn (m>=n) matrix x into an integral matrix with gcd of maximal determinants equal to 1 if p is equal to 0, not divisible by p otherwisenfhermite(nf,x)=if x=[A,I], gives a pseudo-basis of the module sum A_jI_jnorml2(x)=square of the L2-norm of the vector xnucomp(x,y,l)=composite of primitive positive definite quadratic forms x and y using nucomp and nudupl, where l=[|D/4|^(1/4)] is precomputedpfpolint(xa,ya,x)=polynomial interpolation at x according to data vectors xa, yapolredabsfast(x)=a smallest generating polynomial of the number field for the T2 norm on the rootspolsym(x,n)=vector of symmetric powers of the roots of x up to nprimroot(n)=returns a primitive root of n when it existsrayclassno(bnf,x)=ray class number of the module x for the big number field bnf. Faster than buchray if only the ray class number is wantedrnfpolred(nf,pol)=given a pol with coefficients in nf, finds a list of polynomials defining some subfields, hopefully simplersmallbuchinittheta(q,z)=Jacobi sine theta-functionStep 6
(external prettyprint)primelimit[secure mode]: can't modify '%s' default (to %s)denominators not allowed in divisorsZY_ZXY_rnfequationsubresext, dr = %lda3t2incorrect type in %sthis code has to come firstI can't remember before the big bangbe honest for primes from %ld to %ld
FindModulusclass field not totally real in bnrstarkA6gzetakall%c[%ld;%ldmt_POL+ binary output(precp=%ld,valp=%ld):Vecsmall([non zero exponent in gsincoselement_divelement_powsubst: unexpected variable precedencedifferent number of variables and values in substvecincorrect argument in O()not an integer index in jbesselhhyperu's third argument must be positivenstop = %ld
zeta of power seriespolylog around a!=0vecsmall_copyHensel lift... mod p^k (avma - bot = %lu)
non-monic relative polynomialsplotinit(w,{x=0},{y=0},{flag=0}): initialize rectwindow w to size x,y. If flag!=0, x and y express fractions of the size of the current output device. x=0 or y=0 means use the full size of the deviceplotscaler1>15 in discrayabslistarchStarting zidealstarunits computations
       column #%ld of the matrix log(b_j/Q): %Z
factordivexact is not exact!incorrect rectwindow number in graphic function (%ld not in [0, %ld])stroke
Rho: searching small factor of %ld-bit integer
SQUFOF: first cycle exhausted after %ld iterations,
	dropping it
OddPwrs: passed modular checks
IFAC: prime %Z
	appears at least to the power %ld
MPQS: next sort point at %3.1f%%
 %lu %luMPQS: combining
p too small in fixedfieldsympolFixedField: Found: %Z
FixedField: Sym: %Z
GaloisAnalysis:non Galois for p=%ld
MGaloisConj: Fixed field %Z
not a rational polynomial in conjvechnf_special[1]. i=%ldstarting SNF loopgsmithallEVENtoo large precision in preci()No such elliptic curve in databaseRoots: %Z
FFTinitp = 1 in hilbert()discriminant too big in classnoforpari  c11 = %Z
  Entering CF...
LLL failed. Increasing kappa
Not a vector where column of a matrix expectedMath::Pari::lgExpected long return type, got code '%s'gor_gandgaffect (gen_1)additionmultiplicationStr({str}*): concatenates its (string) argument into a single stringaddprimes({x=[]}): add primes in the vector x to the prime table to be used in trial division. x may also be a single integer. Composite "primes" are allowed, and in that case you may later get a message "impossible inverse", which will give you some factors. List the current extra primes if x is omitted. If some primes are added which intersect non trivially the existing table entries, suitable updating is doneGGpagm(x,y): arithmetic-geometric mean of x and yalias("new","old"): new is now an alias for oldasinhbnfissunit(bnf,sfu,x): bnf being output by bnfinit (with flag<=2), sfu by bnfsunit, gives the column vector of exponents of x on the fundamental S-units and the roots of unity if x is a unit, the empty vector otherwiseD1,L,ceilD"",r,D"",s,D0,L,ellchangecurveelljeulerphi(x): Euler's totient function of xGGLD0,L,idealcoprimeidealpow(nf,x,n,{flag=0}): n-th power of the ideal x in HNF in the number field nf If (optional) flag is non-null, reduce the resultimagincgamintfouriersin(X=a,b,x,s,{tab}): numerical integration from a to b of sin(2*Pi*x*X)*s(X) from a to b, where a, b, and tab are as in intnum. This is the sine-Fourier transform if a=-infty and b=+infty.kronecker(x,y): kronecker symbol (x/y)listcreatenfeltreducenfkermodpr(nf,x,pr): kernel of the matrix x in Z_K/pr, where pr is in modpr format (see nfmodprinit)nfmodprinitnfsubfieldsGGDnD0,L,poltchebi(n,{v=x}): Tchebitcheff polynomial of degree n (n C-integer), in variable vqfbcompraw(x,y): Gaussian composition without reduction of the binary quadratic forms x and yrnfeltabstorel(rnf,x): transforms the element x from absolute to relative representationrnfequation(nf,pol,{flag=0}): given a pol with coefficients in nf, gives the absolute equation apol of the number field defined by pol. flag is optional, and can be 0: default, or non-zero, gives [apol,th], where th expresses the root of nf.pol in terms of the root of apolrnfidealreltoabsrnfidealupserlaplacesetrand(n): reset the seed of the random number generator to nteichmullerwrite(filename,a): write the string expression a (same output as print) to filenamewritetex(filename,a): write the string expression a (same format as print) to filename, in TeX formatzeta(s): Riemann zeta function at s with s a complex or a p-adic numbernot enough memory, new stack %luuser interruptgacos     is %Z
non-monic polynomial. Change of variables discardedcodeprime
++++ cglob = %ld: new relation (need %ld)truncation error in bestappr
adjbuchgenfuconductorcoredisc2(n)=(long)gen_2-component row vector [d,f], where d is the discriminant of the quadratic field Q(sqrt(n)) and n=df^2. f may be a half integerdivsum(n,X,expr)=sum of expression expr, X running over the divisors of neta(x)=eta function without the q^(1/24)factoreddiscf(x,p)=discriminant of the maximal order defined by the polynomial x, where p is the matrix of the factorization of the discriminant of xfactornf(x,t)=factorization of the polynomial x over the number field defined by the polynomial thermitehavas(x)=3-component vector [H,U,P] such that H is an (upper triangular) Hermite normal form of x with extra zero columns, U is a unimodular matrix and P is a permutation of the rows such that P applied to xU gives H, using Havas's algorithmidealcoprime(nf,x,y)=gives an element b in nf such that b.x is an integral ideal coprime to the integral ideal yideallistunitisprincipalforce(bnf,x)=same as isprincipal, except that the precision is doubled until the result is obtainedissquare(x)=true(1) if x is a square, false(0) if notjell(x)=elliptic j invariant of xkerintlog(x)=ln(x)=natural logarithm of xnfhermitemod(nf,x,detx)=if x=[A,I], and detx is a multiple of the ideal determinant of x, gives a pseudo-basis of the module sum A_jI_jnfpowoordellpermutation2num(vect)=ordinal (between 1 and n!) of permutation vectpolredabs2(x)=gives [pol,a] where pol is as in polredabs, and alpha is the element whose characteristic polynomial is polreal(x)=real part of xrnfhermitebasis(bnf,order)=given an order as output by rnfpseudobasis, gives either a true HNF basis of the order if it exists, zero otherwisesmalldiscf(x)=discriminant of the number field defined by the polynomial x where one assumes that no square of a prime>primelimit divides the discriminant of xsmallpolredsum(x,X=a,b,expr)=x plus the sum (X goes from a to b) of expression exprtchebizetak(nfz,s)=Dedekind zeta function of the number field nfz at s, where nfz is the vector computed by initzeta (NOT by initalg)[rnfkummer] compositumnaive reduction mod U^l: unit exp. = %Z
[%ld,,%ld]   %s = "%s"
prompt_cont\ifx\%s\undefined
  \def\%s{%s}\fi
polratliftrelatively prime polynomials expectedLLL_cmbf: rank decrease
Time setup: %ld
inverse mod %ld (stable=%ld)empty matrix in supplb2zknot enough flags in string function signaturenot a polynomial of degree 2 in quadrayp = %lu, q = %lu, e = %ld
reverse polmod does not exist+polrelnum = %Z
F_21(7) = 7:3L(7) = L(3,2)t_QFIout of range in integer -> character conversion (%ld)                  
gpow: need integer exponent if series valuation != 0element_mulelement_invmodideal  treating a = %ld, b = %ld
invalid data in hqfevaltwistpartialzeta (2), j = %ld/%ld  3) Final bound: %Z
compositum: %Z inseparablereducible polynomial in allbase content in fastnu is %Z
qpsolubleincorrect idele in idealaddtooneincorrect hash code in decodemoduleStarting discrayabs computations
Default bound for regulator: 0.2
     p divides w(K)
not enough precomputed primes, need primelimit ~ incorrect number of components in rectdrawRho: hang on a second, we got something here...
	found %sfactor = %Z
ifac_moebiusIFAC: factoring %Z
	factor no. %ld is a duplicate%s
Choosing t = %ld
Individual Fermat powerings:
FNEW
MPQS: starting Gauss over F_2 on %ld relations
MPQS: wrapping, more primes for A now chosen near FB[%ld] = %ld
TR_POL(1), i = %ld/%ld2 * Hadamard bound * ind = %Z
lg(Z) = %ld, lg(Y) = %ld
Not a Galois field in a Galois related functionMonomorphismLift: trying early solution %Z
GaloisConj:Frobenius:%Z
S4GaloisConj:Computing isomorphisms %d:%Z
incorrect permutation in permtopol    2nd phase done
[3]: smithalleasycharTerminal does not define linetypenot a t_QFI in powimagpslqget_embCF failed. Increasing kappa
Math::Pari::pari2numMath::Pari::pari_printMath::Pari::typMath::Pari::lgefMath::Pari::initmemToo few args %d for PARI function %sgpowGDGDGD0,L,pvLGagreffepari.psarcherpaddhelp(symbol,"message"): add/change help message for a symbolbigomegabnrrootnumber(bnr,chi,{flag=0}); returns the so-called Artin Root Number, i.e. the constant W appearing in the functional equation of the Hecke L-function associated to chi. Set flag = 1 if the character is known to be primitiveellconvertname(name): convert an elliptic curve name (as found in the elldata database) from a string to a triplet [conductor, isogeny class, index]. It will also convert a triplet back to a curve name.ellpointtozfactorff(x,p,a): factorization of the polynomial x in the finite field F_p[X]/a(X)F_p[X]fibonaccifordiv(n,X,seq): the sequence is evaluated, X running over the divisors of nintnumstep(): gives the default value of m used by all intnum and sumnum  routines, such that the integration step is 1/2^m.matker(x,{flag=0}): basis of the kernel of the matrix x. flag is optional, and may be set to 0: default; non-zero: x is known to have integral entriesnfeltreducemodpr(nf,a,pr): element a modulo pr in nf, where pr is in modpr format (see nfmodprinit)omegapolcoeffpolcoeff(x,s,{v}): coefficient of degree s of x, or the s-th component for vectors or matrices (for which it is simpler to use x[]). With respect to the main variable if v is omitted, with respect to the variable v otherwisepoldiscpolrootsmod(x,p,{flag=0}): roots mod p of the polynomial x. flag is optional, and can be 0: default, or 1: use a naive search, useful for small ppoltschirnhaus(x): random Tschirnhausen transformation of the polynomial xpolylogV=GGEpqfbpowrawquaddiscrnfeltdown(rnf,x): expresses x on the base field if possible; returns an error otherwisesetminus(x,y): set of elements of x not belonging to yiGsimplify(x): simplify the object x as much as possiblesizebyte(x): number of bytes occupied by the complete tree of the object xsolve(X=a,b,expr): real root of expression expr (X between a and b), where expr(a)*expr(b)<=0sumnum(X=a,sig,expr,{tab},{flag=0}): numerical summation of expr from  X = ceiling(a) to +infinity. sig is either a scalar or a two-component vector coding the function's decrease rate at infinity. It is assumed that the scalar part of sig is to the right of all poles of expr. If present, tab must be initialized by sumnuminit. If flag is nonzero, assumes that conj(expr(z)) = expr(conj(z)).thetanullk(q,k): k'th derivative at z=0 of theta(q,z)variableznprimroot(n): returns a primitive root of n when it existsResetting all trapslist = %Z
padicsqrtnlift.degree does not divide phi(n) in subcyclobezoutres(x,y)=gives a 3-dimensional row vector [u,v,d] such that d=resultant(x,y) and u*x+v*y=d, where x and y are polynomialsbinary(x)=gives the vector formed by the binary digits of x (x C-integer)bittest(x,n)=gives bit number n (coefficient of 2^n) of the integer xboundcf(x,lmax)=continued fraction expansion of x with at most lmax termsbuchgenfu(P,...)=compute the structure of the class group, the regulator a primitive root of unity and a system of fundamental units (if they are not too large) for the number field defined by the polynomial P. See manual for the other parameters (which can be omitted)buchinitforcefubuchrayinitbuchrayinit(bnf,ideal)=same as buchrayinitgen, except that the generators are not explicitly computeddeplindirdiv(x,y)=division of the Dirichlet series x by the Dir. series ydireuler(p=a,b,expr)=Dirichlet Euler product of expression expr from p=a to p=b, limited to b terms. Expr should be a polynomial or rational function in p and X, and X is understood to mean p^(-s)idealdiv(nf,x,y)=quotient x/y of two ideals x and y in HNF in the number field nfidealfactor(nf,x)=factorization of the ideal x given in HNF into prime ideals in the number field nfidealhermite2(nf,a,b)=hermite normal form of the ideal aZ_K+bZ_K in the number field K defined by nf, where a and b are elementsideallistarchgenideallistzstar(nf,bound)=vector of vectors of all zidealstarinits of all ideals of norm<=bound, without generatorsintgen(X=a,b,s)=general numerical integration of s from a to b with respect to X, to be used after removing singularitiesintinf(X=a,b,s)=numerical integration of s from a to b with respect to X, where a or b can be plus or minus infinity (1.0e4000), but of same signkerint2(x)=LLL-reduced Z-basis of the kernel of the matrix x with integral entries using a modified LLLlexsort(x)=sort the elements of the vector x in ascending lexicographic orderlindep(x)=Z-linear dependencies between components of x (Hastad et al)lllgrammodpnfmod(nf,a,b)=gives r such that r=a-bq is small with q algebraic integernucomppolredabsfastrayclassnolistredrealrndtoi(x)=take the nearest integer to all the coefficients of x, without taking into account loss of integer part precisionrnfequation(nf,pol)=given a pol with coefficients in nf, gives the absolute equation of the number field defined by polroots(x)=roots of the polynomial x using Schonhage's method modified by Gourdonsetsigmaksignunitsmalldiscfsmallpolred2default: inexistent format\vskip\medskipamount\bgroup\bfunknown default: %sFpX_FpXQV_compo: %d FpXQ_mul [%d]
  alpha_lambda(mu,p) = %Z
    column selection:Q_denomnot a variable:break not allowed in print()can't allow allocatemem() in loopstwo abcissas are equal in polintchk_gen_init: subfield %Z
r  p : %Z - (%Z)
expecting a simple variable in ellwptwo vector/matrix types in bilhellnot a series in serreversetest if polynomial is square-free
  2) Conversion from T_2 --> | |^2 bound : %Z
nf_LLL_cmbf: checking factor %ld (avma - bot = %lu)
plotrpoint(w,dx,dy): draw a point (and move cursor) at position dx,dy relative to present position of the cursor in rectwindow wnfsmith for non square matricesplease apply bnrinit(,,1) and not bnrinit(,,0)bad subgroup in conductor or discraycollecting garbage inimpossible assignment I-->Iimpossibleforbidden[caller of] snextprIFAC: trying MPQS
monomorphismlift()gtracehnfspec[1]mit = %Z
_%ldExpect a number, got a stringLow-level terminals of Gnuplot.  Query available terminals via
	plotterm("??")
Specify size (if it changable via scaling) as in "termname=300,200".
Add terminal options (if applicable) after the name and SPACE char.
See documentation of options in gnuplot, or, if via Term::Gnuplot, via
    perldoc GnuplotTerminals
reducible form in qfr_rhoK%ld conformal_polc14 = %Z
Math::Pari::EpMath::Pari::pari2ivMath::Pari::interface2099Math::Pari::__dumpStackMath::Pari::setseriesprecisionMath::Pari::initprimes$Math::Pari::initprimes not defined!intiter%iarg1, arg2, arg3gcmp0_lexGD0,L,D0,G,RgX_powersprime too big in rootmod2non-positive precision in rootpadicpadicapprpolynomial variable must have higher priority in factorffQpX_to_ZX* Eigenvalues computed
reorderPolrevQfbGpbnrclassnolist(bnf,list): if list is as output by ideallist or similar, gives list of corresponding ray class numbersGDGdilogelllocalredfactorback(f,{e},{nf}): given a factorisation f, gives the factored object back. If this is a prime ideal factorisation you must supply the corresponding number field as last argument. If e is present, f has to be a vector of the same length, and we return the product of the f[i]^e[i]forprimefracgaloispermtopolGD0,L,Dngamma(x): gamma function at xglobalhyperuhyperu(a,b,x): U-confluent hypergeometric functionidealchinese(nf,x,y): x being a prime ideal factorization and y a vector of elements, gives an element b such that v_p(b-y_p)>=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other pidealinv(nf,x,{flag=0}): inverse of the ideal x in the number field nf. If flag is omitted or set to 0, use the different. If flag is 1 do not use itideallistarchintfouriercos(X=a,b,x,s,{tab}): numerical integration from a to b of cos(2*Pi*x*X)*s(X) from a to b, where a, b, and tab are as in intnum. This is the cosine-Fourier transform if a=-infty and b=+infty.intnuminitmathnfmodidmatindexrank(x): gives two extraction vectors (rows and columns) for the matrix x such that the extracted matrix is square of maximal rankmatinverseimagematsolvemodmodreversenfbasisnfeltdivremnfeltmod(nf,a,b): gives r such that r=a-bq is small with q algebraic integernffactor(nf,x): factor polynomial x in number field nfnfgaloisapply(nf,aut,x): Apply the Galois automorphism sigma (polynomial or polymod) to the object x (element or ideal) in the number field nfnfsolvemodprnumdivpadicprec(x,p): absolute p-adic precision of object xpolhenselliftpolroots(x,{flag=0}): complex roots of the polynomial x. flag is optional, and can be 0: default, uses Schonhage's method modified by Gourdon, or 1: uses a modified Newton methodprintpsi(x): psi-function at xqfbclassno(x,{flag=0}): class number of discriminant x using Shanks's method by default. If (optional) flag is set to 1, use Euler productsrandom({N=2^31}): random integer between 0 and N-1rnfequationrnfpolredsetintersect(x,y): intersection of the sets x and ysuminf(X=a,expr): infinite sum (X goes from a to infinity) of real or complex expression exprGD0,L,D1,L,psignificant pointers lost in gerepile! (please report)overflow in real shift########## FACTORBASE ##########

base change =
bestappr/regulatorbasistoalg(nf,x)=transforms the vertical vector x on the integral basis into an algebraic numberbuchinitforcefu(P,...)=compute the necessary data for future use in ideal and unit group computations, and insist on having fundamental units. See manual for detailsbuchinitfu(P,...)=compute the necessary data for future use in ideal and unit group computations, including fundamental units if they are not too large. See manual for detailsbuchrayGDGDGD1,G,conductorofchardet2diagonaldiscf2(x)=discriminant of the number field defined by the polynomial x using round 2factmod(x,p)=factorization mod p of the polynomial x using Berlekamphermitemod(x,d)=(upper triangular) Hermite normal form of x, basis for the lattice formed by the columns of x, where d is the non-zero determinant of this latticeimage2incgam1(s,x)=incomplete gamma function (for debugging only)initell(x)=x being the vector [a1,a2,a3,a4,a6], gives the vector: [a1,a2,a3,a4,a6,b2,b4,b6,b8,c4,c6,delta,j,[e1,e2,e3],w1,w2,eta1,eta2,q,area]label(n)=THIS FUNCTION HAS BEEN SUPPRESSEDlex(x,y)=compare x and y lexicographically (1 if x>y, 0 if x=y, -1 if x<y)lllgramkerimgenlllgramkerimgen(x)=kernel and lll reduction of the lattice whose gram matrix is the matrix x with polynomial coefficientslnlogagm(x)=natural logarithm of x, computed using agm (faster than log for more than a few hundred decimal digits)matextractmodulargcd(x,y)=gcd of the polynomials x and y using the modular methodnfdivres(nf,a,b)=gives [q,r] such that r=a-bq is smallnfmulV=GGIpsimplefactmod(x,p)=same as factmod except that only the degrees of the irreducible factors are givensturmpart(x,a,b)=number of real roots of the polynomial x in the interval (a,b]sumpos(X=a,expr)=sum of positive series expr, the formal variable X starting at athreetotwotrace(x)=trace of xweberf(x)=Weber's f function of x (j=(f^24-16)^3/f^24)wf2\egroup\bgroup\ttremoveprimeZZ_%Z[%Z]/(%Z) is not a field in FpX_ffintersectsubresall, dr = %ld
group:discsr* Time LLL: %ld
* Time Check Factor: %ld
code error in intnumJunk after an id in a stringified flagb6can't replace an existing symbol by an aliasbreak not allowed here (reading arguments)incorrect type or length in matrix assignmentbreak not allowed here (reading long)gen_sortCompute Cl(k)S & TF_18(6):2 = [1/2.S(3)^2]2nfiso or nfinclt_COL
 Top : %lx   Bottom : %lx   Current stack : %lx
 %ld variable names used out of %d

 /%s/%s%s(lg=%ld%s):   next = %s ])	-sqrtrnon positive argument in mplogzero argument in palogmodule too large in ffshanksnon-positive argument in O()lift_interncoset not found in cosets_perm_searchpolynomial variable must have highest priority in nfroots  new order: %Z
  fx = %Z, gx = %Zno root in nilord. Is p = %Z a prime?  entering DecompvGD0,L,V=GGIpD0,M,D0,L,
Parametric|1; Recursive|2; no_Rescale|4; no_X_axis|8; no_Y_axis|16; no_Frame|32; no_Lines|64; Points_too|128; Splines|256; no_X_ticks|512; no_Y_ticks|1024; Same_ticks|2048plothraw(listx,listy,{flag=0}): plot in high resolution points whose x (resp. y) coordinates are in listx (resp. listy). If flag is 1, join points, other non-0 flags should be combinations of bits 8,16,32,64,128,256 meaning the same as for ploth()psplothraw(listx,listy,{flag=0}): same as plothraw, except that the output is a postscript program in psfile (pari.ps by default)unif_mod_fZelement not in ideal in ideal_two_elt2element does not belong to ideal in ideal_two_elt2non Galois extension in rnfnormgroupnot a square matrixreducible polynomialtrying to overwrite a universal objectnot a row vector in plothp = %lu, setting up for B2
composite squfof [caller of] (5n is a square)	- ruled out
IFAC: found %ld large prime (power) factor%s.
found factor
	%Z
currently lost to the factoring machineryRho: time = %6ld ms,	%3ld round%s
...partial impossibly short in ifac_sort_oneMQPS: short of space -- another buffer for sorting
error whilst writing to file %sMPQS: resplitting a factor after %ld kernel vectors
, looking for more...MPQS: wrapping up vector of %ld factors
lifting embedding mod p^k = %Z^%ld
coeff too big for embedding
rootpadicfast()GaloisConj:%Z
GaloisConj: not found, %d hops 
hnf_special[2]. i=%ld    after phase1:
variable must have higher priority in matfrobenius    first pass in hnffinal done
galois files not available
[missing %s]    ----> Group # %ld/%ld:
Incorrect curve name in ellconvertname%s/elldata/ell%lderror setting terminal "%s"Init: trying delta = %d, t = %d
bound = 0 in minim2Z_issquaremissing exponentFp_sqrtincorrect size in pnqnFp_sqrtlidentical index variables in matrixexpected an integer in bnfisintnorminithueratlift$;@Math::Pari::interface29Math::Pari::interface4Math::Pari::interface31Math::Pari::interface35Math::Pari::interface48Math::Pari::interface19Math::Pari::interface86Math::Pari::ifactMath::Pari::have_graphicsg, n, eltin, dummy1, dummy2interfaceV=GIpgtofplistinsertpolrootsmodcan't trap memory errorsaccureruserColVec({x=[]}): transforms the object x into a vector. Empty vector if x is omittedaddhelpalgdep(x,n,{flag=0}): algebraic relations up to degree n of x, using lindep([1,x,...,x^(n-1)], flag).bitandbnfclassunitbnfnarrowbnfsunitconjvec(x): conjugate vector of the algebraic number xellidentify(E): look up the elliptic curve E in the elldata database and return [[N, M, ...], C] where N is the name of the curve in J. E. Cremona database, M the minimal model and C the coordinates change (see ellchangecurve).galoissubfields(G,{flags=0},{v}):Output all the subfields of G. flags have the same meaning as for galoisfixedfieldgcd(x,{y}): greatest common divisor of x and y.getstackglobal(x): declare x to be a global variableidealchineseintfourierexp(X=a,b,x,s,{tab}): numerical integration from a to b of exp(-2*I*Pi*x*X)*s(X) from a to b, where a, b, and tab are as in intnum. This is the ordinary Fourier transform if a=-infty and b=+infty. Note the minus sign.intnumromb(X=a,b,s,{flag=0}): numerical integration of s (smooth in  ]a,b[) from a to b with respect to X. flag is optional and mean 0: default.  s can be evaluated exactly on [a,b]; 1: general function; 2: a or b can be  plus or minus infinity (chosen suitably), but of same sign; 3: s has only  limits at a or bmatfrobeniusmoebiusnewtonpolynfeltmulmodprpolleadpolresultant(x,y,{v},{flag=0}): resultant of the polynomials x and y, with respect to the main variables of x and y if v is omitted, with respect to the variable v otherwise. flag is optional, and can be 0: default, assumes that the polynomials have exact entries (uses the subresultant algorithm), 1 for arbitrary polynomials, using Sylvester's matrix, or 2: using a Ducos's modified subresultant algorithmquadunitrnfidealnormabssigmasizedigitsumnuminittanh(x): hyperbolic tangent of xthetanullkvecextractnon-cyclic case in polsubcyclo: use galoissubcyclo insteadalgdep2(x,n,dec)=algebraic relations up to degree n of x where dec is as in  lindep2apell2discrayabslistdivres(x,y)=euclidean division of x by y giving as a 2-dimensional column vector the quotient and the remainderexp(x)=exponential of xfordiv(n,X,seq)=the sequence is evaluated, X running over the divisors of ngcd(x,y)=greatest common divisor of x and yhermitemodhvectoridealaddmultoneidealval(nf,x,p)=valuation at p given in primedec format of the ideal x in the number field nfinitalgredintnum(X=a,b,s)=numerical integration of s from a to b with respect to Xisirreduciblelll1(x)=old version of lll reduction of the vectors forming the matrix x (gives the unimodular transformation matrix)lllgram1(x)=old version of lll reduction of the lattice whose gram matrix is x (gives the unimodular transformation matrix)order(x)=order of the integermod x in (Z/nZ)*polred(x)=reduction of the polynomial x (gives minimal polynomials only)polzag(n,m)=Zagier's polynomials of index n,mqfi(a,b,c)=binary quadratic form a*x^2+b*x*y+c*y^2 with b^2-4*a*c<0rootpadic(x,p,r)=p-adic roots of the polynomial x to precision rsylvestermatrix(x,y)=forms the sylvester matrix associated to the two polynomials x and y. Warning: the polynomial coefficients are in columns, not in rowstan(x)=tangent of xteich(x)=teichmuller character of p-adic number xweipell(e)=formal expansion in x=z of Weierstrass P functionzidealstarthis function has been suppressed   colors = "%s"
1TeXstyle"/usr/local/lib/pari/gphelp"bitwise orzero argument in an arithmetic functionnot a factorization in polhenselliftnot an integral factorization in polhenselliftleftright_powinexact computation in RgX_extgcdLLL_cmbfDDF: wrong numbers of factorsexponential increase in integral transform[install] identifier '%s' already in usetatebreak not allowed in array contextexpected character: '%c' instead ofFB = %Z
different lengths in polinterpolaten too small (%ld) in numtopermdifferent pointers in ff_poltypegmulsgincorrect data in eltreltoabs.fumain variable in bnrstark must not be xquickpolRecCoeffincorrect prime idealTschirnhaus transform. New pol: %ZF_18(6) = [3^2]2 = 3 wr 2S_4(6d) = [2^2]S(3)matricest_COMPLEXno such function 1)\cr
inputunexpected closing bracenum = coef of degree %ld = agm of two vector/matriceszero argument in mplog[apell1] baby steps, s = %ldisintsum from 1 to N-1p-adic jbessel functionrootexponent: %ld
IndexPartial: discriminantdisc. factorisation0 argument in nfhilbertnfhilbert not soluble at real place %ld
plotrecthraw(w,data,{flags=0}): plot graph(s) for data in rectwindow w, where data is a vector of vectors. If plot is parametric, length of data should be even, and pairs of entries give curves to plot. If not, first entry gives x-coordinate, and the other ones y-coordinates. Admits the same optional flags as plotrecth, save that recursive plot is meaninglesssorry, too many primes to check     Beta list = %Z
impossible inverse modulo: not enough memoryyou must initialize the rectwindow first	B2 = %6lu,	gss = %4ld*420
	(baby step table complete)
Rho: fast forward phase (%ld rounds of 64)...
SQUFOF: found factor %ld^2
compositeIFAC: a factor was a power of another prime factor
misaligned partial detected in ifac_sort_oneNumber of Fermat powerings = %lu
MPQS: done sorting%s, time = %ld ms
MPQS: decomposed a %s
GaloisFixedField:den=%Z mod=%Z 
GaloisConj: Solution too large, discard it.
S4GaloisConj:Testing %d/8 %d:%d:%d
s4test()GaloisIndex: Using hash value s=%ld
hnfspec[2]COSopening %sIncorrect vector in ellconvertnamepositive discriminant in %sqfr_unitPartial = %Z
truncr (precision loss in truncation)Bad PARI variable name "%s" specifiedarg1, arg2, arg3, arg4=0arg1, arg2, arg3, arg4, arg5, arg6=0, arg7=0gpui_gaddUnsupported interface %d and no code for a Pari function %s<null>newvalue = -1forbidden or conflicting type in gvalempty vector in vecmineuclidean division (poldivrem)factormodFqX_split_Trager failed!  ***   
this trap keywordCol({x=[]}): transforms the object x into a column vector. Empty vector if x is omittedSer(x,{v=x}): convert x (usually a vector) into a power series with variable v, starting with the constant coefficientatanbesselh2(nu,x): H^2-bessel function of index nu and argument xbnrconductorofchar(bnr,chi): conductor of the character chi on the ray class group bnrbnrdisc(a1,{a2},{a3},{flag=0}): absolute or relative [N,R1,discf] of the field defined by a1,a2,a3. [a1,{a2},{a3}] is of type [bnr], [bnr,subgroup], [bnf, module] or [bnf,module,subgroup], where bnf is as output by bnfclassunit (with flag<=2), bnr by bnrclass (with flag>0), and subgroup is the HNF matrix of a subgroup of the corresponding ray class group (if omitted, the trivial subgroup). flag is optional whose binary digits mean 1: give relative data; 2: return 0 if module is not the conductorbnrinit(bnf,ideal,{flag=0}): given a big number field as output by bnfinit (only) and an ideal or a 2-component row vector formed by an ideal and a list of R1 zeros or ones representing a module, initializes data linked to the ray class group structure corresponding to this module. flag is optional, and can be 0: default (same as bnrclass with flag = 1), 1: compute also the generators (same as bnrclass with flag = 2). The fifth component is the ray class group structurebnrstark(bnr,{subgroup}): bnr being as output by bnrinit(,,1), finds a relative equation for the class field corresponding to the module in bnr and the given congruence subgroup (the trivial subgroup if omitted) using Stark's units. The ground field and the class field must be totally real.centerliftdirdiv(x,y): division of the Dirichlet series x by the Dirichlet series yellorderfactorialffinit(p,n,{v=x}): monic irreducible polynomial of degree n over F_p[v]idealnormidealval(nf,x,p): valuation at p given in idealprimedec format of the ideal x in the number field nfisfundamental(x): true(1) if x is a fundamental discriminant (including 1), false(0) if notispseudoprime(x,{n}): true(1) if x is a strong pseudoprime, false(0) if not. If n is 0 or omitted, use BPSW test, otherwise use strong Rabin-Miller test for n randomly chosen basesissquare(x,{&n}): true(1) if x is a square, false(0) if not. If n is given puts the exact square root there if it was computedmatdetintmatfrobenius(M,{flag},{v=x}): Return the Frobenius form of the square matrix M. If flag is 1, return only the elementary divisors as a vector of polynomials in the variable v. If flag is 2, return a two-components vector [F,B] where F is the Frobenius form and B is the basis change so that M=B^-1*F*B.mathnf(A,{flag=0}): (upper triangular) Hermite normal form of A, basis for the lattice formed by the columns of A. flag is optional whose value range from 0 to 4 (0 if omitted), meaning : 0: naive algorithm. 1: Use Batut's algorithm. Output 2-component vector [H,U] such that H is the HNF of A, and U is a unimodular matrix such that AU=H. 3: Use Batut's algorithm. Output [H,U,P] where P is a permutation matrix such that P A U = H. 4: as 1, using a heuristic variant of LLL reduction along the waymathnfmod(x,d): (upper triangular) Hermite normal form of x, basis for the lattice formed by the columns of x, where d is a multiple of the non-zero determinant of this latticematrixqzmatsize(x): number of rows and columns of the vector/matrix x as a 2-vectorminpoly(A,{v=x}): minimal polynomial of the matrix or polmod A.nfeltdivGGGGnfeltval(nf,a,pr): valuation of element a at the prime pr as output by idealprimedecnfisincl(x,y): tests whether the number field x is isomorphic to a subfield of y (where x and y are either polynomials or number fields as output by nfinit). Return 0 if not, and otherwise all the isomorphisms. If y is a number field, a faster algorithm is usedpolredabs(x,{flag=0}): a smallest generating polynomial of the number field for the T2 norm on the roots, with smallest index for the minimal T2 norm. flag is optional, whose binary digit mean 1: give the element whose characteristic polynomial is the given polynomial. 4: give all polynomials of minimal T2 norm (give only one of P(x) and P(-x)). 16: partial reductionquadclassunit(D,{flag=0},{tech=[]}): compute the structure of the class group and the regulator of the quadratic field of discriminant D. If flag is non-null (and D>0), compute the narrow class group. See manual for the optional technical parametersreadvec({filename}): create a vector whose components are the evaluation of all the expressions found in the input file filenamernfconductorrnfidealhnfrnfinitrnfisnorm(T,x,{flag=0}): T is as output by rnfisnorminit applied to L/K. Tries to tell whether x is a norm from L/K. Returns a vector [a,b] where x=Norm(a)*b. Looks for a solution which is a S-integer, with S a list of places in K containing the ramified primes, generators of the class group of ext, as well as those primes dividing x. If L/K is Galois, omit flag, otherwise it is used to add more places to S: all the places above the primes p <= flag (resp. p | flag) if flag > 0 (resp. flag < 0). The answer is guaranteed (i.e x norm iff b=1) if L/K is Galois or, under GRH, if S contains all primes less than 12.log(disc(M))^2, where M is the normal closure of L/Ksinsqrtint(x): integer square root of x (x integer)sqrtn(x,n,{&z}): nth-root of x, n must be integer. If present, z is set to a suitable root of unity to recover all solutions. If it was not possible, z is set to zerosumdivthueinitbroken pipeBernoulli sum
#### Computing fundamental units
Computing powers for subFB: %Z

#### Tentative regulator : %Z
Subcyclo: prime l=%Z
buchfu(bnf)=compute the fundamental units of the number field bnf output by buchinitbuchnarrowchar(x,y)=det(y*I-x)=characteristic polynomial of the matrix x using the comatrixchell(x,y)=change data on elliptic curve according to y=[u,r,s,t]compimag(x,y)=Gaussian composition of the binary quadratic forms x and y of negative discriminantcompositum(pol1,pol2)=vector of all possible compositums of the number fields defined by the polynomials pol1 and pol2core2cosh(x)=hyperbolic cosine of xfactoredpolred(x,p)=reduction of the polynomial x, where p is the matrix of the factorization of the discriminant of x (gives minimal polynomials only)frac(x)=fractional part of x=x-floor(x)hermitehavasjbesselh(n,x)=J-bessel function of index n+1/2 and argument x, where n is a non-negative integerlegendrelllkerimmatextract(x,y,z)=extraction of the components of the matrix x according to the vector or masks y (for the rows) and z (for the columns) from left to right (1,2,4,8,...for the first, second, third, fourth, ...rows or columns)minim2(x,bound)=looks for vectors of square norm <= bound, return the first one and its normperfpermutation2numrnflllgram(nf,pol,order)=given a pol with coefficients in nf and an order as output by rnfpseudobasis or similar, gives [[neworder],U], where neworder is a reduced order and U is the unimodular transformation matrixsignatsmallpolred(x)=partial reduction of the polynomial x (gives minimal polynomials only)tchebi(n)=Tchebitcheff polynomial of degree n (n C-integer)threetotwo(nf,a,b,c)=returns a 3-component vector [d,e,U] such that U is a unimodular 3x3 matrix with algebraic integer coefficients such that [a,b,c]*U=[0,d,e]lightbg[1,,1], [5,,1], [3,,1], [7,,1], [6,,1], , [2,,1]compatiblestrictmatchdifferent variables in modulargcdgcd mod %lu (bound 2^%ld)not a correct factorization in polhenselliftgengroupbreak not allowed after ^local() bloc must appear before any other expressiona 0x0 matrix has no elementsincorrect lextype in vecsortRecpolnumI/O: leaked file descriptor (%d): %sLINES,CLONEint = pol = mod =  - 0.E%ldE%ldTeX variable name too longsqrtnrlog is not meromorphic at 0gcotanzero element in zsigneforbidden substitution by a vectorpolevalpoleval: i = %ldjbesselp-adic kbessel functionincorrect variables in rnfcharpolybound computationimpossible inverse: %Z  fastnu: HNF(G) is computed
install(name,code,{gpname},{lib}): load from dynamic library 'lib' the function 'name'. Assign to it the name 'gpname' in this GP session, with argument code 'code'. If 'lib' is omitted use 'libpari.so'. If 'gpname' is omitted, use 'name'plotcursorplotpointtypecannot invert zero ideal     p divides h(K)
Warning: increasing precnot an integer argument in an arithmetic functiondivision by zeroIFAC: checking for pure square
IFAC: factor %Z
	is %s
IFAC: a factor was divisible by another prime factor,
	leaving a cofactor = %Z
MPQS: found multiplier %ld for N
ftell error on full relations filecannot seek FREL fileNumberOfConjugates:Nbtest=%ld,card=%ld,p=%ld
GaloisFixedField:cosets=%Z 
i = %ld: %Z
	%2ld: %Z
GNUPLOT_DRAW_DLLGNUPLOT_DRAW_DLL_NO_PERLauto/Term/Gnuplot/not an integer exponent in nupowdiscriminant not congruent to 0,1 mod 4 in primeformd must be 0 or 3 mod 4Candidate: %Z
negative accuracy in lindep2refine_H  - norm sol. no %ld/%ld
Not enough precision in thueColumns of input matrix are of different heightMath::Pari::interface109$$$$Math::Pari::type_namegdivvLLGDVDExnegative length in listcreateZp_apprkilling bloc (no %ld): %08lx
pari.logOagmbitor(x,y): bitwise "or" of two integers x and y. Negative numbers behave as if modulo big power of 2bnfcertify(bnf): certify the correctness (i.e. remove the GRH) of the bnf data output by bnfclassunit or bnfinitbnfisintnorm(bnf,x): compute a complete system of solutions (modulo units of positive norm) of the absolute norm equation N(a)=x, where a belongs to the maximal order of big number field bnf (if bnf is not certified, this depends on GRH)bnrclassnolistbnrdisccos(x): cosine of xdivrem(x,y,{v}): euclidean division of x by y giving as a 2-dimensional column vector the quotient and the remainder, with respect to v (to main variable if v is omitted)ellchangepoint(x,y): change data on point or vector of points x on an elliptic curve according to y=[u,r,s,t]ellpoweval(x): evaluation of x, replacing variables by their valueforprime(X=a,b,seq): the sequence is evaluated, X running over the primes between a and bgaloispermtopol(gal,perm): gal being a galois field as output by galoisinit and perm a element of gal.group, return the polynomial defining the corresponding Galois automorphismgetstack(): current value of stack pointer avmaintnuminitgenispseudoprimelift(x,{v}): lifts every element of Z/nZ to Z or T[x]/PT[x] to T[x] for a type T if v is omitted, otherwise lift only polymods with main variable v. If v does not occur in x, lift only intmodslog(x): natural logarithm of x.matimagecomplGGDVDVDImatsnfnfhnfpolsym(x,n): vector of symmetric powers of the roots of x up to nprecisionprodinfquadray(D,f,{lambda}): relative equation for the ray class field of conductor f for the quadratic field of discriminant D (which can also be a bnf). For D < 0, lambda (if supplied) is the technical element of bnf  necessary for Schertz's method. In that case, return 0 if lambda is not suitable.rnfidealnormabs(rnf,x): absolute norm of the ideal xrnfidealup(rnf,x): lifts the ideal x (of the base field) to the relative fieldrnfpolred(nf,pol): given a pol with coefficients in nf, finds a list of relative polynomials defining some subfields, hopefully simplersqr(x): square of x. NOT identical to x*xsumnuminit(sig, {m=0}, {sgn=1}): initialize tables for numerical summation. sgn is 1 (in fact >= 0), the default, for sumnum (ordinary sums)  or -1 (in fact < 0) for sumnumalt (alternating sums). sig is as in sumnum and m is as in intnuminit.trace(x): trace of xtruncate(x,{&e}): truncation of x; when x is a power series,take away the O(X^). If e is present, do not take into account loss of integer part precision, and set e = error estimate in bitswrite1segmentation fault: bug in PARI or calling programnon-positive integer argument in cxpsigetfusub factorbase (%ld elements)Subcyclo: orbits=%Z
generators must be prime to conductor in galoissubcyclobuchfubuchray(bnf,ideal)=given a big number field as output by buchinitfu (only) and  an ideal or a 2-component row vector formed by an ideal and a list of R1 zeros or ones representing a module, finds the ray class group structure corresponding to this modulebytesize(x)=number of bytes occupied by the complete tree of the object xconjvec(x)=conjugate vector of the algebraic number xcore(n)=unique (positive of negative) squarefree integer d dividing n such that n/d is a squarederiv(x,y)=derivative of x with respect to the main variable of yfactfqfactoredbasis(x,p)=integral basis of the maximal order defined by the polynomial x, where p is the matrix of the factorization of the discriminant of xfactoreddiscffpn(p,n)=monic irreducible polynomial of degree n over F_p[x]intersectisfund(x)=true(1) if x is a fundamental discriminant (including 1), false(0) if notisisomfastlaplace(x)=replaces the power series sum of a_n*x^n/n! by sum of a_n*x^npermutationprecision(x)=real precision of object xprimrootprincipalideleV=GID0,L,pranksturmpartteichvecmax(x)=maximum of the elements of the vector/matrix xPARIpromptSTARTtimerdatadirpanic: set_optimizeaddprimecan't accept 0 in addprimesresultant mod %ld (bound 2^%ld, stable=%ld)subgroup    mu = polhensellift: factors %Z and %Z are not coprimepseudodiv dx = %ld >= %ldfactorsintnuminit0incorrect abscissa in sumnumincorrect table length in intnum initializationnegative size in matid_FlmCannot negate id=value in a stringified flagcan't pop gp variablefujpositive integer expected[quadhilbert] incorrect values in pq: %ludoubling stack in dirmul
different modulus in ff_poltypebase field not totally real in bnrstarkpolrel = %Z
incorrect idealgpolcomp (different degrees)D(4)F(5) = 5:4LLL basisMat(%s is not a GP binary fileYvaluation of j must be negative in p-adic ellinitelement_mulizidealij done
this object is a leaf. It has no componentsinvalid data in qfevalgtopolyjbessel around a!=0... lifted (avma - bot = %lu)
Ideals to consider:
non monic relative equationplotlines(w,listx,listy,{flag=0}): draws an open polygon in rectwindow w where listx and listy contain the x (resp. y) coordinates of the vertices. If listx and listy are both single values (i.e not vectors), draw the corresponding line (and move cursor). If (optional) flag is non-zero, close the polygonplotrmoveV=GGIpD0,L,D0,L,
PHASE 2: are all primes good ?

M0 = %Z
error opening overflow in R->dbl conversionToo few points (%ld) for spline plotMiller-Rabin: testing base %ld
snextpr: %lu != prc210_rp[%ld] mod 210
IFAC: unfactored composite declared primeMPQS: primes for A to be chosen near FB[%ld] = %ld
	%Z%s
comp.1, 0, vandermondeinversemod()%d.%Zgnormallhnfmod[1]. i=%ld    Output of isin_%ld_G_H(%ld,%ld): not included.
compositionmid = %ZFincke-Pohst, final LLL: prec = %ld
cauchy_boundx1 -> %Z
Checking (\pm %Z, \pm %Z)
invmodrtodblVariable in sv2pari is not of known typeMath::Pari::interface10Math::Pari::interface15Math::Pari::interface12$;$$$;$$$Math::Pari::interface49Math::Pari::interface44Math::Pari::pari_texprintgdiventggt_gbitorvV=GGGgtolongkernelfrobeniusspec_FpXQ_pow%lu.: %ld
Euler=Euler(): Euler's constant with current precisionStrchr(x): converts x to a string, translating each integer into a characterbernfracbinarybinomial(x,y): binomial coefficient x*(x-1)...*(x-y+1)/y! defined for y in Z and any xbitnegimply(x,y): bitwise "negated imply" of two integers x and y, in other words, x BITAND BITNEG(y). Negative numbers behave as if modulo big power of 2bnfcertifybreakceil(x): ceiling of x=smallest integer>=xGDnD0,L,coredireuler(p=a,b,expr,{c}): Dirichlet Euler product of expression expr from p=a to p=b, limited to b terms. Expr should be a polynomial or rational function in p and X, and X is understood to mean p^(-s). If c is present, output only the first c termsellztopointeta(x,{flag=0}): if flag=0, eta function without the q^(1/24), otherwise eta of the complex number x in the upper half plane intelligently computed using SL(2,Z) transformationsfactorpadic(x,p,r,{flag=0}): p-adic factorization of the polynomial x to precision r. flag is optional and may be set to 0 (use round 4) or 1 (use Buchmann-Lenstra)for(X=a,b,seq): the sequence is evaluated, X going from a up to bvV=GGGIforsubgroup(H=G,{bound},seq): execute seq for each subgroup H of the abelian group G (in SNF form), whose index is bounded by bound. H is given as a left divisor of G in HNF formgaloisisabeliangetrandhilbert(x,y,{p}): Hilbert symbol at p of x,y.ideallogissquarefreelcmmatidmatsnf(x,{flag=0}): Smith normal form (i.e. elementary divisors) of the matrix x, expressed as a vector d. Binary digits of flag mean 1: returns [u,v,d] where d=u*x*v, otherwise only the diagonal d is returned, 2: allow polynomial entries, otherwise assume x is integral, 4: removes all information corresponding to entries equal to 1 in dnfgaloisconj(nf,{flag=0},{den}): list of conjugates of a root of the polynomial x=nf.pol in the same number field. flag is optional (set to 0 by default), meaning 0: use combination of flag 4 and 1, always complete; 1: use nfroots; 2 : use complex numbers, LLL on integral basis (not always complete); 4: use Allombert's algorithm, complete if the field is Galois of degree <= 35 (see manual for detail). nf can be simply a polynomial with flag 0,2 and 4, meaning: 0: use combination of flag 4 and 2, not always complete (but a warning is issued when the list is not proven complete); 2 & 4: same meaning and restrictions. Note that only flag 4 can be applied to fields of large degrees (approx. >= 20)norml2poldegree(x,{v}): degree of the polynomial or rational function x with respect to main variable if v is omitted, with respect to v otherwise. For scalar x, return 0 is x is non-zero and a negative number otherwiseprime(n): returns the n-th prime (n C-integer)print1D"",s,reorder({x=[]}): reorder the variables for output according to the vector x. If x is void or omitted, print the current list of variablesrnfbasistoalgrnfeltdownrnfinit(nf,pol): pol being a non constant irreducible polynomial defined over the number field nf, initializes a vector of data necessary for working in relative number fields (rnf functions). See manual for technical detailsrnfsteinitzserreverse(x): reversion of the power series xsign(x): sign of x, of type integer, real or fractionsubstvecthueinit(P,{flag=0}): initialize the tnf corresponding to P, that will be used to solve Thue equations P(x,y) = some-integer. If flag is non-zero, certify the result unconditionnaly. Otherwise, assume GRH (much faster of course)whilewritebin(filename,{x}): write x as a binary object to file filename. If x is omitted, write all session variablesbad component %ld in object %Zargument too large in ggamdggammalngamma around a!=1not a factorization matrix in isunitbnfnewpreccompute_Rapprpadic(x,a)=p-adic roots of the polynomial x congruent to a mod pchelldetintdirzetak(nf,b)=Dirichlet series of the Dedekind zeta function of the number field nf up to the bound b-1disc(x)=discriminant of the polynomial xfactgauss(a,b)=gaussian solution of ax=b (a matrix,b vector)hilbidealinv(nf,x)=inverse of the ideal x in the number field nf not using the differentideallistarchgen(nf,list,arch)=vector of vectors of all zidealstarinits of all modules in list with archimedean arch added, with generatorsidealmul(nf,x,y)=product of the two ideals x and y in the number field nfincgam(s,x)=incomplete gamma functionintersect(x,y)=intersection of the vector spaces whose bases are the columns of x and yV=GGID1,L,pisprincipalray(bnf,x)=bnf being output by buchrayinit, gives the vector of exponents on the ray class group generators of x. In particular x is principal if and only if the result is the zero vectorlllintpartial(x)=partial (hence faster) lll reduction of the vectors forming the matrix x when the gram matrix is integral (gives the unimodular transformation matrix)lllratmatrixqz3minidealmodp(x,y)=creates the integer x modulo y as a permanent object (on the heap)principalideal(nf,x)=returns the principal ideal generated by the algebraic number x in the number field nfqfrredrealnodrndtoirootssign(x)=sign of x, of type integer, real or fractionsmallbuchinit(pol)=small buchinit, which can be converted to a big one using makebigbnfsumalt(X=a,expr)=Villegas-Zagier's acceleration of alternating series expr, X starting at atwototwo(nf,a,b)=returns a 3-component vector [d,e,U] such that U is a unimodular 2x2 matrix with algebraic integer coefficients such that [a,b]*U=[d,e] and d,e are hopefully smallerStep 8
beta reduced = %Z
\ifx\%s\undefined
  \def\%s#1#2{%s}\fi
bitwise negated implyIFAC: (Partial fact.) Initial stop requested.
factor_irred_matnot an integral polynomial in polhenselliftNaive recombinationinexact computation in subresextpolsturm, dr = %ldm too large in intnuminitincorrect vector in matmuldiagonalregproduct, error bits = %ldquadhilbert (pq)incorrect data in findquadray regulatorCompute polrelnumincorrect nf in nfnewprect_REALt_RFRACList((lgeflist=%ld):imag = /var/tmpgpow: modulus %Z is not primegsinnot an integer exponent in nfpowarch_to_permnf_Pohlig-Hellman: DL mod %Z^%ld
gfloorperm_to_GAPempty group in group_domainsquarefree testsplitting mod %Znot a pseudo-basis in nfsimplifybasis  (Fa, Ea) = (%ld,%ld)
  ** switching to normal mode
can't find symbol '%s' in library '%s'plotcopyplotstringpsploth(X=a,b,expr,{flags=0},{n=0}): same as ploth, except that the output is a postscript program in psfile (pari.ps by default)bug2 in nfsmith0 in get_arch_real   BOUND = %ld
unexpected charactersorry, not yet available on this systeminconsistent data in rectplothinECM: %lu should have been prime but isn't
SQUFOF: square form (%ld^2, %ld, %ld) on first cycle
	after %ld iterations, time = %ld ms
ifac_numdivMPQS: found factors = %Z
	and %Z
MQPS: line wrap -- another buffer for sorting
 %ld %ldMPQS: combined %ld full relation%s
Chosen prime: p = %ld
priority of optional variable too high in galoisfixedfieldGaloisConj: exp %d: s=%ld [%ld] a=%ld w=%ld wg=%ld sr=%ld
%d gconj    1st phase done
allhnfmod[2]. i=%ld%s group
(Terminal does not define vectornon invertible polynomial in Flxq_invclassno with too small order%Z eliminated because of sign
Math::Pari::interface_flexible_gen%d unused args for PARI function %sExpected GEN return type, got code '%s'_gbitshiftl_gorvGVgaffect (gnil)negative index (%ld) in listputno more room in this L (size %ld)FpXQYQ_powfactorpadicroots2not a prime in polrootsmod[FqX_split] splitting time: %ld (%ld trials)
to_Fq in %s.GGLD0,L,pGGD1,L,pbnfmakeconj(x): the algebraic conjugate of xdefaultdirzetak(nf,b): Dirichlet series of the Dedekind zeta function of the number field nf up to the bound b-1ellglobalredellisoncurveerfceulerphifloornfalgtobasisnfeltpownfmodprinit(nf,pr): transform the 5 element row vector pr representing a prime ideal into modpr format necessary for all operations mod pr in the number field nf (see manual for details about the format)nfrootsof1polhensellift(x, y, p, e): lift the factorization y of x modulo p to a factorization modulo p^e using Hensel lift. The factors in y must be pairwise relatively prime modulo pGDGDGD&polredabspolsympolzagierqfbcomprawqfbnupow(x,n): n-th power of primitive positive definite quadratic form x using nucomp and nuduplGD0,L,DGDGDGqfgaussredqflllquadhilbertquadhilbert(D,{pq}): relative equation for the Hilbert class field of the quadratic field of discriminant D (which can also be a bnf). If D<0, pq (if supplied) is a 2-component vector [p,q], where p,q are the prime numbers needed for Schertz's method. In that case, return 0 if [p,q] not suitable.rnfalgtobasis(rnf,x): relative version of nfalgtobasis, where rnf is a relative numberfieldrnfbasis(bnf,order): given an order as output by rnfpseudobasis or rnfsteinitz, gives either a basis of the order if it is free, or an n+1-element generating setsubgrouplist(bnr,{bound},{flag=0}): bnr being as output by bnrinit or a list of cyclic components of a finite Abelian group G, outputs the list of subgroups of G (of index bounded by bound, if not omitted), given as HNF left divisors of the SNF matrix corresponding to G. If flag=0 (default) and bnr is as output by bnrinit, gives only the subgroups for which the modulus is the conductorV=GGEDGD0,L,pznprimrootgashsingular argument in atanhpowFBgenbe_honest() failure on prime %Z
not a HNF matrix in galoissubcycloN must be a bnrinit or a znstar if H is a matrix in galoissubcycloaddprimes(x)=add primes in the vector x (with at most 20 components) to the prime tableclassno(x)=class number of discriminant xdirmul(x,y)=multiplication of the Dirichlet series x by the Dir. series ygetstack()=current value of stack pointer avmahess(x)=Hessenberg form of xintegmakebigbnfmatrixqz2(x)=finds a basis of the intersection with Z^n of the lattice spanned by the columns of xqfismithtransvector(n,X,expr)=row vector with n components of expression expr (X ranges from 1 to n)helpFpXQ_sqrtnmatrix cyclo|gtranspanicc6unused charactersnarrow class groupnot a set in setsearchincorrect subgrp in bnrstarkAllStarkCiktempfile %s already existsthqfi(close pipecan't expand ~/tmplocalred (p | c6)apell (f^(i*s) = 1)gpolvarnon-real argument in eint1PermutationGroup<1|>number of factor(s) found: %ld
rnfdet2ordmaxplotcolorvLLGGD0,L,D0,L,plotpoints(w,listx,listy): draws in rectwindow w the points whose x (resp y) coordinates are in listx (resp listy). If listx and listy are both single values (i.e not vectors), draw the corresponding point (and move cursor)non-integral exponent in idealpowred[2]: discrayabslistarchprecision too lowincorrect typepostscriptOddPwrs: testing for exponent %ld
ifac_issquarefreeIFAC: main loop: %ld factor%s left
IFAC: repeated factor %Z
	detected in ifac_sort_one
MPQS: creating factor base and allocating arrays...
MPQS: sieve threshold = %u
LPTMP
MPQS: time in Gauss and gcds = %ld ms
,%lu7th powerBest lift: %d
gnorml1sqred1H = %Z
C = %Z
[1]: smithall i = %ldPermutation: %Z
Galois names files not available, please upgrade galdata
[missing %s]not a real quadratic form in redreallllfp (exact)gcd f_P  does not divide n_p
Pari.cMath::Pari::interface2Math::Pari::interface26$$$$$Math::Pari::PARIvarpanic: PARI narg value not attachedgDid not get a variablearg1, arg2=0, arg3=0, arg4=0gegalDo not know how to work with Pari control structure `%s'%sstack size is %ld bytes (%d x %ld longs)
gaffect (geuler)listkill     real coefficients polynomial, using zrhqr()

For full compatibility with GP 1.39.15, type "default(compatible,3)", or set "compatible = 3" in your GPRC fileSerasinbinary(x): gives the vector formed by the binary digits of x (x integer)bitneg(x,{n=-1}): bitwise negation of an integers x truncated to n bits. n=-1 means represent infinite sequences of bit 1 as negative numbers. Negative numbers behave as if modulo big power of 2bitxorbnfisnormbnrL1(bnr, {subgroup}, {flag=0}): bnr being output by bnrinit(,,1) and subgroup being a square matrix defining a congruence subgroup of bnr (the trivial subgroup if omitted), for each character of bnr trivial on this subgroup, compute L(1, chi) (or equivalently the first non-zero term c(chi) of the expansion at s = 0). The binary digits of flag mean 1: if 0 then compute the term c(chi) and return [r(chi), c(chi)] where r(chi) is the order of L(s, chi) at s = 0, or if 1 then compute the value at s = 1 (and in this case, only for non-trivial characters), 2: if 0 then compute the value of the primitive L-function associated to chi, if 1 then compute the value of the L-function L_S(s, chi) where S is the set of places dividing the modulus of bnr (and the infinite places), 3: return also the charactersbnrconductorcontent(x): gcd of all the components of x, when this makes sensecosderiv(x,{y}): derivative of x with respect to the main variable of y, or to the main variable of x if y is omittedellakellheightmatrix(e,x): gives the height matrix for vector of points x on elliptic curve e using theta functionslincgamcmatadjointmatcompanion(x): companion matrix to polynomial xmatindexrankmatkerint(x,{flag=0}): LLL-reduced Z-basis of the kernel of the matrix x with integral entries. flag is optional, and may be set to 0: default, uses a modified LLL, 1: uses matrixqznfbasistoalg(nf,x): transforms the column vector x on the integral basis into an algebraic numbernfdetint(nf,x): multiple of the ideal determinant of the pseudo generating set xnfeltpow(nf,a,k): element a^k in nfnfnewprec(nf): transform the number field data nf into new data using the current (usually larger) precisionpolcycloLLDnqfbhclassnoreadvecremoveprimessetunion(x,y): union of the sets x and ysimplifysumaltvectorisprincipal (incompatible bnf generators)
#### Looking for %ld relations (small norms)
relation cancelled: D = %Z
den = %Z
be honestaddellassmat(x)=associated matrix to polynomial xGfcompositum2decodefactor(fa)=given a factorisation fa, gives the factored object backdegreediagonal(x)=creates the diagonal matrix whose diagonal entries are the entries of the vector xdiscf2GD0,G,D0,G,D2,L,divisors(x)=gives a vector formed by the divisors of x in increasing orderGGffactoredpolred2goto(n)=THIS FUNCTION HAS BEEN SUPPRESSEDhermite2(x)=2-component vector [H,U] such that H is an (upper triangular) Hermite normal form of x, basis for the lattice formed by the columns of x, and U is a unimodular matrix such that xU=H, using Batut's algorithmidealdivexactidealhermite(nf,x)=hermite normal form of the ideal x in the number field nf, whatever form x may haveideallistunitarchgen(bnf,lists,arch)=adds the archimedean arch to the lists output by ideallistunitgenideallllredidealmulredidealtwoelt2(nf,x,a)=(long)gen_2-element representation of an ideal x in the number field nf, with the first element equal to aincgam2initalg(x)=x being a nonconstant irreducible polynomial, gives the vector: [x,[r1,r2],discf,index,[M,MC,T2,T,different] (see manual),r1+r2 first roots, integral basis, matrix of power basis in terms of integral basis, multiplication table of basis]isprime(x)=true(1) if x is a strong pseudoprime for 10 random bases, false(0) if notisprincipalrayissqfreekbessellambdaklllgramgenpolylogdold(m,x)=D_m-modified m-th polylog of xrnfpseudobasis(nf,pol)=given a pol with coefficients in nf, gives a 4-component vector [A,I,D,d] where [A,I] is a pseudo basis of the maximal order in HNF on the power basis, D is the relative ideal discriminant, and d is the relative discriminant in nf^*/nf*^2rootmod2(x,p)=roots mod p of the polynomial x, when p is smallrootsof1sin(x)=sine of xtanh(x)=hyperbolic tangent of xtschirnhaus(x)=random Tschirnhausen transformation of the polynomial x[rnfkummer] conductorStep 16
realprecisiondebugmem(prettymatrix)(prettyprint)colors   %s = %lu
Too large primelimitDegree list for ERS (trials: %ld) = %Z
pol[Frobenius]DDF_roots, m = %ldQ_div_to_intnon-invertible polynomial in RgXQ_inv* [mod p]poltufu, sorry, couldn't deal with this field. PLEASE REPORTquadhilbertimagrnfinitalgincorrect bigidealD(5) = 5:2D(7) = 7:2a(i,j)t_SERt_VECt_STRunknown type %ld\cdotI can't see into the futureNULL
%ld%s column =  \left(gtanmpsc1can't compute tan(Pi/2 + kPi)not a prime in apellw1 and w2 R-linearly dependent in elliptic functionincompatible variables in algtobasisnot the same number field in rnfalgtobasisPermutationGroup<%3ld %s at prime
  %Z
Time: %ld
absolute basisincorrect coeff in rnf functionvL       prime ideal Q: %Z
not enough precomputed primesrectlines) %ld %ld m 90 rotate show -90 rotate
ECM: working on %ld curves at a time; initializing	(got initial helix)
IFAC: factor %Z
	is prime (no larger composite)
Maximal number of nondeterministic steps = %lu
MPQSMPQS: found factor = %Z

MPQS: restarting sieving ...
	(estimated memory needed: %4.1fMBy)
MPQS: renamed file %s to %s
[3]: mpqs_solve_linear_systemp = %ld,	lcm = %ld,	orbits: %Z

***** Entering subfields

pol = %Z
NumberOfConjugates:card=%ld,p=%ld
MonomorphismLift: lift to prec %dgnorml2Terminal does not define optionsDISPLAYBuffer overflow finding gnuplot DLLget_term_ftablebitsize bound: %ld
qzer[%ld]=%ld
bound too largecomposite modulus in Fp_sqrt: %Zcontfrac0  c13 = %Z
SmallSolssol = %Z
Math::Pari::is_gnilMath::Pari::installPerlFunctionCVMath::Pari::interface9900inPari.xsGD0,L,DGpvV=GED0,L,GD0,G,D0,G,D0,L,p %2ld: %s
listconcatpolynomial has probably multiple roots in zrhqr/usr/local/lib/pari/Cannot initialize kernelGDnPol(x,{v=x}): convert x (usually a vector or a power series) into a polynomial with variable v, starting with the leading coefficientSetLbesselkbnfinitbnfmake(sbnf): transforms small sbnf as output by bnfinit with flag=3 into a true big bnfconjdirzetakellminimalmodelellsubelltors(e,{flag=0}): torsion subgroup of elliptic curve e: order, structure, generators. If flag = 0, use Doud's algorithm; if flag = 1, use Lutz-NagellGDGD0,L,pPerrorfactorgaloisidentifygaloissubcyclo(N,H,{fl=0},{v}):Compute a polynomial (in variable v) defining the subfield of Q(zeta_n) fixed by the subgroup H of (Z/nZ)*. N can be an integer n, znstar(n) or bnrinit(bnfinit(y),[n,[1]],1). H can be given by a generator, a set of generator given by a vector or a HNF matrix (see manual). If flag is 1, output only the conductor of the abelian extension. If flag is 2 output [pol,f] where pol is the polynomial and f the conductor.gammahidealadd(nf,x,y): sum of two ideals x and y in the number field defined by nfidealprincipalV=GGGEDGpV=GGED0,L,D0,L,pisprimevGmatdet(x,{flag=0}): determinant of the matrix x using Gauss-Bareiss. If (optional) flag is set to 1, use classical gaussian elimination (slightly better for integer entries)mathilbertmathilbert(n): Hilbert matrix of order n (n C-integer)mathnfmodmatimage(x,{flag=0}): basis of the image of the matrix x. flag is optional and can be set to 0 or 1, corresponding to two different algorithmsmax(x,y): maximum of x and ynfbasistoalgnfeltdivmodprnfeltdivmodpr(nf,a,b,pr): element a/b modulo pr in nf, where pr is in modpr format (see nfmodprinit)nfeltmulmodpr(nf,a,b,pr): element a. b modulo pr in nf, where pr is in modpr format (see nfmodprinit)nfeltreduce(nf,a,id): gives r such that a-r is in the ideal id and r is smallnfeltreducemodprnfinit(pol,{flag=0}): pol being a nonconstant irreducible polynomial, gives the vector: [pol,[r1,r2],discf,index,[M,MC,T2,T,different] (see manual),r1+r2 first roots, integral basis, matrix of power basis in terms of integral basis, multiplication table of basis]. flag is optional and can be set to 0: default; 1: do not compute different; 2: first use polred to find a simpler polynomial; 3: outputs a two-element vector [nf,Mod(a,P)], where nf is as in 2 and Mod(a,P) is a polymod equal to Mod(x,pol) and P=nf.pol; 4: as 2 but use a partial polred; 5: is to 3 what 4 is to 2nfsolvemodpr(nf,a,b,pr): solution of a*x=b in Z_K/pr, where a is a matrix and b a column vector, and where pr is in modpr format (see nfmodprinit)polgalois(x): Galois group of the polynomial x (see manual for group coding). Return [n, s, k, name] where n is the order, s the signature, k the index and name is the GAP4 name of the transitive group.polrecip(x): reciprocal polynomial of xpolsylvestermatrixprodinf(X=a,expr,{flag=0}): infinite product (X goes from a to infinity) of real or complex expression. flag can be 0 (default) or 1, in which case compute the product of the 1+expr insteadquaddisc(x): discriminant of the quadratic field Q(sqrt(x))quadpoly(D,{v=x}): quadratic polynomial corresponding to the discriminant D, in variable vquadregulatorrnfconductor(bnf,polrel,{flag=0}): conductor of the Abelian extension of bnf defined by polrel. The result is [conductor,rayclassgroup,subgroup], where conductor is the conductor itself, rayclassgroup the structure of the corresponding full ray class group, and subgroup the HNF defining the norm group (Artin or Takagi group) on the given generators rayclassgroup[3]. If flag is non-zero, check that polrel indeed defines an Abelian extensionserreverseshiftsinh(x): hyperbolic sine of xvecmax(x): maximum of the elements of the vector/matrix xvecmin(x): minimum of the elements of the vector/matrix xvector(n,{X},{expr=0}): row vector with n components of expression expr (X ranges from 1 to n). By default, fill with 0szero argument in gargp-adic lngamma functionlim, nn: [%ld, %ld]
isunitincorrect parameters in classgroupbnr must be over Q in bnr_to_znstarSubcyclo: elements:wrong modulus in galoissubcycloasinh(x)=inverse hyperbolic sine of xboundfactbuchinit(P,...)=compute the necessary data for future use in ideal and unit group computations. See manual for detailscf2(b,x)=continued fraction expansion of x (x rational,real or rational function), where b is the vector of numerators of the continued fractionchar1coeff(x,s)=coefficient of degree s of x, or the s-th component for vectors or matrices (for which it is simpler to use x[])factpol2gamma(x)=gamma function at xgaussmodulo2(M,D,Y)=all solutions of system of congruences MX=Y mod Dideallistzstaridealtwoelt(nf,x)=(long)gen_2-element representation of an ideal x in the number field nfinverseimageisoncurveisprincipalgen(bnf,x)=bnf being output by buchinit, gives [v,alpha,bitaccuracy], where v is the vector of exponents on the class group generators and alpha is the generator of the resulting principal ideal. In particular x is principal if and only if v is the zero vectorkaramullift(x)=lifts every element of Z/nZ to Z or Z[x]/PZ[x] to Z[x]ln(x)=log(x)=natural logarithm of xlngamma(x)=logarithm of the gamma function of xnfdetint(nf,x)=multiple of the ideal determinant of the pseudo generating set xnfsmithnfsmith(nf,x)=if x=[A,I,J], outputs [c_1,...c_n] Smith normal form of xorderorderell(e,p)=order of the point p on the elliptic curve e over Q, 0 if non-torsionpipolvarrecip(x)=reciprocal polynomial of xsmallinitell(x)=x being the vector [a1,a2,a3,a4,a6], gives the vector: [a1,a2,a3,a4,a6,b2,b4,b6,b8,c4,c6,delta,j]smith2sqredsturmzellnot an n-th power in idealsqrtn[secure mode]: Do you want to modify the 'secure' flag? (^C if not)
tex2mail -TeX -noindent -ragged -by_parFpX_resultant (da = %ld)coeff bound: %Z^%ld
RecombinationQ_muli_to_intresultantducos, degpol Q = %ldnon-monic polynomial in poldiscreducedTotal Time: %ld
===========
nfshift operand too biginteger too bigtruc(): n = %ldlocalformal derivationquadhilbertimag (can't find p,q)not a set in setunionincorrect character in bnrrootnumberthe ground field must be distinct from QArtinNumberLooking for a modulus of norm: Compute Wzetavalues = %Z
* conductor no %ld/%ld (N = %ld)
	Init: A4S5F_36(6) = 1/2[S(3)^2]22S_4(6) = [2^3]S(3) = 2 wr S(3)galois (bug4)round4i = %ld
precision too low in chk_gen_initt_LISTrun-away comment. Closing itcouldn't find a suitable name for a tempfile (%s)rd(%c,varn=%ld):[secure mode]: about to write to '%s'. OK ? (^C if not)
gpow: nth-root does not exist1/0 exponent in gsqrtnp-adic argument out of range in gcospadic_sqrtelement_invnot a matrix in matalgtobasis%Z not a nfeltinv_serground%Z^%ld
  Increasing Fa
plotstring(w,x,{flags=0}): draw in rectwindow w the string corresponding to x. Bits 1 and 2 of flag regulate horizontal alignment: left if 0, right if 2, center if 1. Bits 4 and 8 regulate vertical alignment: bottom if 0, top if 8, v-center if 4. Can insert additional gap between point and string: horizontal if bit 16 is set, vertical if bit 32 is setnot a matrix of maximal rank in nfhermite0th power in idealpowprime_spec  Testing primes | h(K)


Starting bnrclassno computations
sorry,False prime number %Z in plisprime	found factors = %Z, %Z,
	and %Z
ifac_bigomega	yielded `factor' %Z
	which isn't!
MPQS: number to factor N = %Z
MPQS: largest prime in FB = %ld

MPQS: passing the %3.1f%% sort point, time = %ld ms
MPQS: no factors found.
error whilst flushing file %slongerFixedField: Size: %ldx%ld
Polynomial not squarefree in galoisinitA4GaloisConj:sigma=%Z 
matrixqz0discriminant = %Z
$$$$$ New prec = %ld
dumbnot a t_QFI in nuduplnegative number of vectors in minim0smallvectorszero modulus in znprimroottoo many iterations in solvePARI: %.*s%*s%.*s%*s%sPerl function exported into PARI did not return a value$Math::Pari::interface20Math::Pari::interface28_oldMath::Pari::interface47Math::Pari::interface59Math::Pari::memUsageMath::Pari::PARI_DEBUGg, narg1, arg2, invgmod_gmul_gpuiUnsupported Pari function %s, interface 0 code NULLBerlekamp_matrixfactorffnot a prime in factmodassignmentvSsGGD1,L,bnfisunit(bnf,x): bnf being output by bnfinit (with flag<=2), gives the column vector of exponents of x on the fundamental units and the roots of unity if x is a unit, the empty vector otherwisechinesecontfracpnqndirmuleint1ellanelltorsforell(E,a,b,seq): execute seq for each elliptic curves E of conductor between a and b in the elldata database.galoissubgroups(G):Output all the subgroups of Gif(a,seq1,seq2): if a is nonzero, seq1 is evaluated, otherwise seq2. seq1 and seq2 are optional, and if seq2 is omitted, the preceding comma can be omitted alsointfouriersinissquarefree(x): true(1) if x is squarefree, false(0) if notkill(x): kills the present value of the variable or function x. Returns new value or 0matalgtobasis(nf,x): nfalgtobasis applied to every element of the matrix xmatbasistoalg(nf,x): nfbasistoalg applied to every element of the matrix xmatinverseimage(x,y): an element of the inverse image of the vector y by the matrix x if one exists, the empty vector otherwisenfeltdiveucnffactormodnfgaloisconjnfisisompolred(x,{flag=0},{p}): reduction of the polynomial x (gives minimal polynomials only). Second and third args are optional. The following binary digits of flag are significant 1: partial reduction, 2: gives also elements. p, if present, contains the complete factorization matrix of the discriminantqfbsolve(Q,p): Return [x,y] so that Q(x,y)=p where Q is a binary quadratic form and p a prime number, or 0 if there is no solution.rnfkummer(bnr,{subgroup},{deg=0}): bnr being as output by bnrinit, finds a relative equation for the class field corresponding to the module in bnr and the given congruence subgroup (the ray class field if subgroup is omitted). deg can be zero (default), or positive, and in this case the output is the list of all relative equations of degree deg for the given bnrrnfsteinitz(nf,order): given an order as output by rnfpseudobasis, gives [A,I,D,d] where (A,I) is a pseudo basis where all the ideals except perhaps the last are trivialGLptypewhile(a,seq): while a is nonzero evaluate the expression sequence seq. Otherwise 0writebingarggthgachBernoulli  Testing P = %Z
isprincipalnot a vector/matrix in cleanarchacosh(x)=inverse hyperbolic cosine of xbigomega(x)=number of repeated prime divisors of xdecodemodule(nf,fa)=given a coded module fa as in discrayabslist, gives the true moduledenomGD0,G,D0,G,D1,L,eval(x)=evaluation of x, replacing variables by their valueglobalred(e)=e being an elliptic curve, returns [N,[u,r,s,t],c], where N is the conductor of e, [u,r,s,t] leads to the standard model for e, and c is the product of the local Tamagawa numbers c_pi=i()=square root of -1idealintersect(nf,x,y)=intersection of two ideals x and y in HNF in the number field defined by nfideallistzstargen(nf,bound)=vector of vectors of all zidealstarinits of all ideals of norm<=bound, with generatorsV=GGID2,L,pisisomlcm(x,y)=least common multiple of x and y=x*y/gcd(x,y)nfdivresordredpermutation(n,k)=permutation number k (mod n!) of n letters (n C-integer)phipnqnpolypolyrev(x,v)=convert x (usually a vector or a power series) into a polynomial with variable v, starting with the constant termprodinf1quaddisc(x)=discriminant of the quadratic field Q(sqrt(x))recipredimagsetminus(x,y)=set of elements of x not belonging to ysturm(x)=number of real roots of the polynomial xStep 12
Step 14, 15 and 17
%ld(%ld) seriesprecision1, 5, 3, 7, 6, 2, 3user functions re-initializedPARIoutpathdefault: incorrect value for %s [%lu-%lu]n-th prime meaningless if n = %ldbad degrees in FpX_ffintersect: %d,%d,%dFpV_polintFinal lambda = %ld
forsubgroup (alpha != countsub)...tried prime %3ld (%-3ld %s). Time = %ld
incorrect range in extractFpM_gauss_pivotrenaming a GP variable is forbidden) = break not allowed here (defining global var)[none]unfinished stringgeneratorsinvalid bound in randomnormalizing a series with 0 leading termNot enough precomputed primes (need all p <= %ld)please apply nfinit firstincorrect matrix for idealpolred for non-monic polynomial  generator: %Z
chk_gen_init: new prec = %ld (initially %ld)
t_PADIC + setting %s
p^l : :
   hash = %3ld, valence = %3ld, menu = %2ld, code = %s
undefined environment variable: %s%s%sbadwith a = %Z
gceilvectosmallt_SER with negative valuation in gtopolyinv_szeta_euler, p = %lu/%lutrueeta)  Increasing Ea
primedec: %Z is not primezpsolubleqpsolublenfplotclipplotcopy(sourcew,destw,dx,dy,{flag=0}): copy the contents of rectwindow sourcew to rectwindow destw with offset (dx,dy). If flag's bit 1 is set, dx and dy express fractions of the size of the current output device, otherwise dx and dy are in pixels. dx and dy are relative positions of northwest corners if other bits of flag vanish, otherwise of: 2: southwest, 4: southeast, 6: northeast cornersplotrlineplotrpoint[2]: nfhermitemod, i = %ld(\)	(got [p]Q, p = %lu = prc210_rp[%ld] mod 210)
SQUFOF: square form (%ld^2, %ld, %ld) on second cycle
	after %ld iterations, time = %ld ms
IFAC: found %Z =
	%Z ^2
IFAC: forcing ECM, may take some time
  %-3ld: %3ld
MPQS: coefficients A will be built from %ld primes each
%s :%s
FREL file truncated?!coeff too big for pol g(x)
Calcul polynomesS4GaloisConj:Testing %d/3:%d/4:%d/4:%d/4:%Z
not a positive definite matrix in sqred1incompatible matrices in hnf_specialhnfadd (%ld + %ld)extendedgcdhnfpermhnfall[1], li = %ldhnffinal, i = %ld%s/galdata/%s%ld_%ld_%ldmore than %ld rational integer roots
Elliptic files %s not compatible

Recomputing Gram-Schmidt, kmax = %ld
non integral index in sumnegative number of rows in matrixnon integral index in sumpos2prec = %d
  Entering LLL...
The longword %ld ordinal out of boundMath::Pari::converted$$;$$$XSUB call through interface did not provide *functiongnegl <= 2 in greffecvtopnot a prime in rootmodto_Fq_pol.
 %s is not yet implemented.divisionQfb(a,b,c,{D=0.}): binary quadratic form a*x^2+b*x*y+c*y^2. D is optional (0.0 by default) and initializes Shanks's distance if b^2-4*a*c>0s*Strchrbernreal(x): Bernoulli number B_x, as a real number with the current precisionbesselh2binomialbnfunit(bnf): compute the fundamental units of the number field bnf output by bnfinit when they have not yet been computed (i.e. with flag=2)bnrdisclist(bnf,bound,{arch}): gives list of discriminants of ray class fields of all conductors up to norm bound, in a long vector The ramified Archimedean places are given by arch; all possible values are taken if arch is omitted. Supports the alternative syntax bnrdisclist(bnf,list), where list is as output by ideallist or ideallistarch (with units)bnrisprincipalGDGD0,L,coredisc(n,{flag=0}): discriminant of the quadratic field Q(sqrt(n)). If (optional) flag is non-null, output a two-component row vector [d,f], where d is the discriminant of the quadratic field Q(sqrt(n)) and n=df^2. f may be a half integerellwp(e,{z=x},{flag=0}): Complex value of Weierstrass P function at z on the lattice generated over Z by e=[om1,om2] (e as given by ellinit is also accepted). Optional flag means 0 (default), compute only P(z), 1 compute [P(z),P'(z)], 2 consider om as an elliptic curve and compute P(z) for that curve (identical to ellztopoint in that case). If z is omitted or is a simple variable, return formal expansion in zfactorbackfactormod(x,p,{flag=0}): factorization mod p of the polynomial x using Berlekamp. flag is optional, and can be 0: default or 1: simple factormod, same except that only the degrees of the irreducible factors are givenffinitidealappr(nf,x,{flag=0}): x being a fractional ideal, gives an element b such that v_p(b)=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other p. If (optional) flag is non-null x must be a prime ideal factorization with possibly zero exponentsintmellininv(X=sig,x,s,{tab}): numerical integration on the  line real(z) = sig (or sig[1]) of s(z)x^(-z)dz/(2*I*Pi), i.e. inverse Mellin  transform of s at x. sig is coded as follows: either it is real, and then by default assume s(z) decreases like exp(-z). Or sig = [sigR, al], sigR is the abcissa of integration, and al = 0 for slowly decreasing functions, or al > 0 if s(z) decreases like exp(-al*z). tab is as in intnum. Use  intmellininvshort if several values must be computed.liftmatadjoint(x): adjoint matrix of xmatbasistoalgmateigenmathess(x): Hessenberg form of xmatpascal(n,{q}): Pascal triangle of order n if q is omited. q-Pascal triangle otherwisematrix(m,n,{X},{Y},{expr=0}): mXn matrix of expression expr, the row variable X going from 1 to m and the column variable Y going from 1 to n. By default, fill with 0snewtonpoly(x,p): Newton polygon of polynomial x with respect to the prime ppolcompositum(pol1,pol2,{flag=0}): vector of all possible compositums of the number fields defined by the polynomials pol1 and pol2. If (optional) flag is set (i.e non-null), output for each compositum, not only the compositum polynomial pol, but a vector [pol,al1,al2,k] where al1 (resp. al2) is a root of pol1 (resp. pol2) expressed as a polynomial modulo pol, and a small integer k such that al2+k*al1 is the chosen root of polpollegendrepolsturmpoltschirnhausV=GED0,L,ppsiquadrayrnfidealmulrnfpseudobasis(nf,pol): given a pol with coefficients in nf, gives a 4-component vector [A,I,D,d] where [A,I] is a pseudo basis of the maximal order in HNF on the power basis, D is the relative ideal discriminant, and d is the relative discriminant in nf^*/nf*^2setisset(x): true(1) if x is a set (row vector with strictly increasing entries), false(0) if notsumsumdiv(n,X,expr): sum of expression expr, X running over the divisors of nzetakzetakinit(x): compute number field information necessary to use zetak, where x is an irreducible polynomialzncoppersmith(P, N, X, {B=N}): finds all integers x0 with |x0| <= X such that  gcd(N, P(x0)) > B. X should be smaller than exp((log B)^2 / (deg(P) log N)).bad object %Zgatannon-positive integer argument in ggammaclassgroupallv[%ld]=%.4g degree <= 0 in galoissubcycloakell(e,n)=computes the n-th Fourier coefficient of the L-function of the elliptic curve eapellcompimagconj(x)=the algebraic conjugate of xcvtoidiscrayabslistarchalldiscrayabslistlong(bnf,bound)=gives list of discrayabscond of all modules up to norm bound without archimedean places, in a longvector formatgalois(x)=Galois group of the polynomial x (see manual for group coding)gaussmodulo2hellhermitemodid(x,d)=(upper triangular) Hermite normal form of x concatenated with d times the identity matrixidealaddone(nf,x,y)=when the sum of two ideals x and y in the number field K defined by nf is equal to Z_K, gives a two-component vector [a,b] such that a is in x, b is in y and a+b=1idealinv2(nf,x)=inverse of the ideal x in the number field nf using the differentindexrank(x)=gives two extraction vectors (rows and columns) for the matrix x such that the exracted matrix is square of maximal rankinteg(x,y)=formal integration of x with respect to the main variable of yintopenisideal(nf,x)=true(1) if x is an ideal in the number field nf, false(0) if notisprincipalgenforceisprincipalgenforce(bnf,x)=same as isprincipalgen, except that the precision is doubled until the result is obtainedissetkerilength(x)=number of non code words in xlllgramgen(x)=lll reduction of the lattice whose gram matrix is x with polynomial coefficients (gives the unimodular transformation matrix)max(x,y)=maximum of x and yordell(e,x)=y-coordinates corresponding to x-ordinate x on elliptic curve epi=pi()=the constant pi, with current precisionpointellpolred2(x)=reduction of the polynomial x (gives elements and minimal polynomials)polredabsall(x)=complete list of the smallest generating polynomials of the number field for the T2 norm on the rootspolvar(x)=main variable of object x. Gives p for p-adic x, error for scalarsprimedecrootmod2subell(e,z1,z2)=difference of the points z1 and z2 on elliptic curve esupplementtaniyamatorsellunitvaluation(x,p)=valuation of x with respect to pzidealstar(nf,I)=3-component vector v, giving the structure of (Z_K/I)^*. v[1] is  the order (i.e. phi(I)), v[2] is a vector of cyclic components, and v[3]  is a vector giving the corresponding generatorsStep 9, 10 and 11
\egroup   %s = %lu %s
factor_irred
found factor %Z
remaining modular factor(s): %ld
%s already exists with incompatible valenceunknown functionnot a proper member definitionincorrect vector or matrixbreak not allowed in assignmentcan't modify a pre-defined member: ...need %ld more relations
computeP2not a vector in permtonumnot a polmod in modreverseincorrect subgroup in bnrL1S&TA5C(7) = 7S7non-monic polynomial. Result of the form [nf,c]chk_gen_init: difficult field, trying random elements
%c[0mt_POLMODt_VECSMALLthis object uses debugging variablescould not open requested file %sI/O: checking output pipe...
localred (nu_D - nu_j != 0,6)leaving
forbidden substitution by a non square matrixinvalid quadratic form in qfbevalinvalid quadratic form in hqfevalp-adic jbesselh functionkbessel around a!=0wrong argument in galoisisabelian
Entering nffactor:
Using Trager's method
nf_LLL_cmbfplothplotlinesnored + denominator in idealapprfact       generator of (Zk/Q)^*: %Z
%10s%-9.7g%*.7g
single vector in gtodblList%ld %ld m %ld %ld l %ld %ld l %ld %ld l closepath
ECM: time = %6ld ms
ECM: dsn = %2ld,	B1 = %4lu,ECM: time = %6ld ms,	p <= %6lu,
	found factor = %Z
Rho: time = %6ld ms,	Pollard-Brent giving up.
SQUFOF: ...but the root form seems to be on the principal cycle
 3rd%s	   %3ld:  %3ld   (3rd %ld, 5th %ld, 7th %ld)
miller(rabin)IFAC: main loop: this was the last factor
`washere' out of bounds in ifac_sort_oneMPQS: number too big to be factored with MPQS,
	giving upLPNEWCOMBMPQS: found %lu candidate%s
MPQS: done sorting one file.
	packaging %ld: %Z ^%ld (%s)
cube
***** Leaving subfields

galoisconj2polsqred3hnfspec[3], (i,j) = %ld,%ldincorrect value in bin()Incorrect curve name in ellsearch/blib/libarg to partition function must be < 10^15lllfpnot a p-adic vector in plindephigher degree than expected in algdepdiscriminant too large in classno  c6  = %Z
Math::Pari::interface0Math::Pari::interface2091Math::Pari::interface22geqGD0,L,Dgaffect (ghalf)%s fileabs(x): absolute value (or modulus) of xatanhbesselj(nu,x): J-bessel function of index nu and argument xbezoutGGbitnegimplydenominatorellan(e,n): computes the first n Fourier coefficients of the L-function of the elliptic curve e (n<2^24 on a 32-bit machine)elleta(om): om=[om1,om2], returns the two-component row vector [eta1,eta2] of quasi-periods associated to [om1,om2]ellinit(x,{flag=0}): x being the vector [a1,a2,a3,a4,a6] defining the curve Y^2 + a1.XY + a3.Y = X^3 + a2.X^2 + a4.X + a6, gives the vector: [a1,a2,a3,a4,a6,b2,b4,b6,b8,c4,c6,disc,j,[e1,e2,e3],w1,w2,eta1,eta2,area]. If the curve is defined over a p-adic field, the last six components are replaced by root,u^2,u,q,w,0. If optional flag is 1, omit them altogether. x can also be a string, in this case the coefficients of the curve with matching name are looked in the elldata database if available.ellpow(e,x,n): n times the point x on elliptic curve e (n in Z)lGDGgettime(): time (in milliseconds) since last call to gettimeGLD4,L,matimageGD0,L,DGpolsturm(x,{a},{b}): number of real roots of the polynomial x in the interval]a,b] (which are respectively taken to be -oo or +oo when omitted)polsubcyclo(n,d,{v=x}): finds an equation (in variable v) for the d-th degree subfields of Q(zeta_n). Output is a polynomial or a vector of polynomials is there are several such fields, or none.print(a): outputs a (in raw format) ending with newlineprint1(a): outputs a (in raw format) without ending with newlineqfbprimeformrnfidealnormrelrnflllgramrnfpolredabs(nf,pol,{flag=0}): given a pol with coefficients in nf, finds a relative simpler polynomial defining the same field. Binary digits of flag mean: 1: return also the element whose characteristic polynomial is the given polynomial, 2: return an absolute polynomial, 16: partial reductionsubstsum(X=a,b,expr,{x=0}): x plus the sum (X goes from a to b) of expression exprtheta(q,z): Jacobi sine theta-functionD"",r,DIDIvecsort(x,{k},{flag=0}): sorts the vector of vectors (or matrix) x in ascending order, according to the value of its k-th component if k is not omitted. Binary digits of flag (if present) mean: 1: indirect sorting, return the permutation instead of the permuted vector, 2: sort using lexicographic order, 4: use descending instead of ascending orderggamdgamd of a power seriesBernoullisEnd of PHASE 1.

bnfinit: %s%s, not givenfor this ideal
m = %Z
znstar_conductoralgtobasis(nf,x)=transforms the algebraic number x into a column vector on the integral basis nf[7]buchinitGD0,G,D0.1,G,D0.1,G,D5,G,pfactfq(x,p,a)=factorization of the polynomial x in the finite field F_p[X]/a(X)F_p[X]gettime()=time (in milliseconds) since last call to gettimehell2(e,x)=canonical height of point x on elliptic curve E defined by the vector e computed using Tate's methodhermitemodididealadd(nf,x,y)=sum of two ideals x and y in the number field defined by nfidealaddoneidealhermite2idmat(n)=identity matrix of order n (n C-integer)intopen(X=a,b,s)=numerical integration of s from a to b with respect to X, where s has only limits at a or bisprincipal(bnf,x)=bnf being output by buchinit, gives the vector of exponents on the class group generators of x. In particular x is principal if and only if the result is the zero vectorkeri(x)=basis of the kernel of the matrix x with integer entrieslllnextprime(x)=smallest prime number>=xnfvalordred(x)=reduction of the polynomial x, staying in the same orderprodeuler(X=a,b,expr)=Euler product (X runs over the primes between a and b) of real or complex expressionprodinf1(X=a,expr)=infinite product (X goes from a to infinity) of real or complex 1+expressionStep 1
reducing beta = %Z
1, 6, 3, 4, 5, 2, 3(on)I was expecting an integer herebound for resultant coeffs: 2^%ld
    lambda =   subgroup:Hensel lift (mod %Z^%ld)nfrootsQintegral transformZM_inv donegauss_pivot_kerarray index (%ld) out of allowed range lambda = %Z
different variables in convolnot an invertible dirseries in dirdivcurve not defined over a p-adic fieldno non-trivial character in bnrL1CplxModulusS2A7 mod \pmatrix{ run-away string. Closing itcouldn't find a suitable name for a tempdir (%s)COLUMNSgsincos  z  = %Z
  z2 = %Z
orderell for nonrational elliptic curvestorsell (bug1)not the same number field in basistoalgnon positive valuation in a series substitutiongtruncargument equal to one in zetachoice of a prime idealIndexPartial: factorizationpolcompositum0  beta = %Z
  ** switching to fast mode
nilordnfhilbertpplotbox(w,x2,y2): if the cursor is at position (x1,y1), draw a box with diagonal (x1,y1) and (x2,y2) in rectwindow w (cursor does not move)plotclip(w): clip the contents of the rectwindow to the bounding box (except strings)plotrecth(w,X=xmin,xmax,expr,{flags=0},{n=0}): plot graph(s) for expr in rectwindow w, where expr is scalar for a single non-parametric plot, and a vector otherwise. If plotting is parametric, its length should be even and pairs of entries give points coordinates. If not, all entries but the first are y-coordinates. Both flags and n are optional. Binary digits of flags mean: 1 parametric plot, 2 recursive plot, 4 do not rescale w, 8 omit x-axis, 16 omit y-axis, 32 omit frame, 64 do not join points, 128 plot both lines and points. n specifies the number of reference points on the graph (0=use default value). Returns a vector for the bounding boxpsdraw(list, {flag=0}): same as plotdraw, except that the output is a postscript program in psfile (pari.ps by default), and flag!=0 scales the plot from size of the current output device to the standard postscript plotting sizenfhermite, i = %ldnot an Abelian extension in rnfnormgroupnon-positive bound in Discrayabslist       new rank: %ld
Warning: failed tobug inthe PARI stack overflows !exponent (expo) overflowstroke showpage
%ld %ld m
PL: proving primality of N = %Z
IFAC: (Partial fact.)Stop requested.
[caller of] elladd0	factor no. %ld was unique%s
IFAC: prime %Z
	appears with exponent = %ld

Step5: testing conditions lp
FREL; sorry, too many block systems in nfsubfieldspolynomial not in Z[X] in galoisconj4galoisborne()conjugates list may be incomplete in nfgaloisconjGaloisConj: B=%Z
A4GaloisConj:I will test %ld permutations
GaloisConj:next p=%ld
GaloisConj:p=%ld deg=%ld fp=%ld

    Reordering of the roots: Terminal does not define movelllint_marked
Checking LLL basis...lllfp[2], kmax = %ldcomposite modulus in Fl_sqrt: %lunegative nmax in sfcontsfcontconstant term != 1 in direulerSemirat. reduction: B0 -> %Z
thue (totally rational case)ratlift: must have 2*amax*bmax < m, found
	amax=%Z
	bmax=%Z
	m=%Z
v5.26.0$$$Math::Pari::PARImatMath::Pari::interface27$$$;$Math::Pari::interface37Math::Pari::interface83Math::Pari::dumpStackarg1, arg2, arg3, arg4, arg0=0arg1, arg2, arg3, arg4, arg5stack size is %ld bytes (%d x %ld longs)
gtolistRgX_RgXQ_compotoo many iterations in rootsold(): using roots2()
polished roots = %Zvariable out of range in reorder  ###   user error: Matbesseli(nu,x): I-bessel function of index nu and argument xbesselk(nu,x,{flag=0}): K-bessel function of index nu and argument x (x positive real of type real, nu of any scalar type). flag is optional, and may be set to 0: default; 1: use hyperubitorGDGDGD0,L,changevar(x,y): change variables of x according to the vector yellbilellbil(e,z1,z2): canonical bilinear form for the points z1,z2 on the elliptic curve e. Either z1 or z2 can also be a vector/matrix of pointsellminimalmodel(e,{&v}): return the standard minimal integral model of the rational elliptic curve e. Sets v to the corresponding change of variableserfc(x): complementary error functionerror("msg"): abort script with error message msgfactornf(x,t): factorization of the polynomial x over the number field defined by the polynomial tvV=GGIvVLLIforvecgettimeidealcoprime(nf,x,y): gives an element b in nf such that b. x is an integral ideal coprime to the integral ideal yideallistideleprincipal(nf,x): returns the principal idele generated by the algebraic number x in the number field nfifispowerlength(x): number of non code words in x, number of characters for a stringlindeplngamma(x): logarithm of the gamma function of xmatintersectnfinitnumbpartomega(x): number of distinct prime divisors of xpoldegreeqfgaussred(x): square reduction of the (symmetric) matrix x (returns a square matrix whose i-th diagonal term is the coefficient of the i-th square in which the coefficient of the i-th variable is 1)qfrepqfsign(x): signature of the symmetric matrix xrnfcharpolyGGGDnrnfdetrnfdiscrnfnormgroup(bnr,polrel): norm group (or Artin or Takagi group) corresponding to the Abelian extension of bnr.bnf defined by polrel, where the module corresponding to bnr is assumed to be a multiple of the conductor. The result is the HNF defining the norm group on the given generators in bnr[5][3]shift(x,n): shift x left n bits if n>=0, right -n bits if n<0.GVEthuetrap({err}, {rec}, {seq}): try to execute seq, trapping error err (all of them if err ommitted); sequence rec is executed if the error occurs and is the result of the command. When seq is omitted, define rec as a default handler for error err (a break loop will be started if rec omitted). If rec is the empty string "" pop out the last default handlerunknown signal**** Testing Different = %Z
    is %Z
initalg & rootsof1looking hard for %Z
algtobasisbasis(x)=integral basis of the field Q[a], where a is a root of the polynomial x, using the round 4 algorithmbernreal(x)=Bernoulli number B_x, as a real number with the current precisionbuchimag(D,...)=compute the structure of the class group of the complex quadratic field of discriminant D<0. See manual for the other parameters (which can be omitted)det2(x)=determinant of the matrix x (better for integer entries)discrayabslistlongdiscrayrelcondeulerfactcantorfor(X=a,b,seq)=the sequence is evaluated, X going from a up to bhessidealchinese(nf,x,y)=x being a prime ideal factorization and y a vector of elements, gives an element b such that v_p(b-y_p)>=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other pideallist(nf,bound)=vector of vectors of all ideals of norm<=bound in nfindsortinitalgred2intinfisprincipalforceisunit(bnf,x)=bnf being output by buchinit, gives the vector of exponents of x on the fundamental units and the roots of unity if x is a unit, the empty vector otherwisekerint1(x)=LLL-reduced Z-basis of the kernel of the matrix x with rational entries using matrixqz3 and the HNFlambdak(nfz,s)=Dedekind lambda function of the number field nfz at s, where nfz is the vector computed by initzeta (NOT by initalg)lllgram(x)=lll reduction of the lattice whose gram matrix is x (gives the unimodular transformation matrix)lllgram1lllgramkerim(x)=kernel and lll reduction of the lattice whose gram matrix is the integral matrix xlllkerimgenmakebigbnf(sbnf)=transforms small sbnf as output by smallbuchinit into a true big bnfminim2munfreducepnqn(x)=[p_n,p_{n-1};q_n,q_{n-1}] corresponding to the continued fraction xpolred2powrealrawqfr(a,b,c,d)=binary quadratic form a*x^2+b*x*y+c*y^2 with b^2-4*a*c>0 and distance dredimag(x)=reduction of the binary quadratic form x with D<0redreal(x)=reduction of the binary quadratic form x with D>0round(x)=take the nearest integer to all the coefficients of xsmallbasis(x)=integral basis of the field Q[a], where a is a root of the polynomial x where one assumes that no square of a prime>primelimit divides the discriminant of xsmith2(x)=gives a three element vector [u,v,d] where u and v are square unimodular matrices such that d=u*x*v=diagonal(smith(x))sqr(x)=square of x. NOT identical to x*xsqrt(x)=square root of xznstar(n)=3-component vector v, giving the structure of (Z/nZ)^*. v[1] is  the order (i.e. phi(n)), v[2] is a vector of cyclic components, and v[3]  is a vector giving the corresponding generatorsbeta reduced via ell-th root = %Z
debugfilesPARIinputEND_contprompttoo many divisors (more than %ld)zero argument in factorintFpXQ_sqrtlFpXQ_matrix_powsZX_resultantFF l-Gen:next %Z
pseudorem dx = %ld >= %ld  alpha = %Z
polsym of a negative nnot a prime number in polhenselliftbuilding treeLLL_cmbf: checking factor %ld
splitting mod p = %ldincorrect a or b in intnummask too large in vecextractbnfbreak not allowed in O()quadray: looking for [a,b] != unit mod 2f
[a,b] = No, we're done!
Modulus = %Z and subgroup = %Z
Checking the square-root of the Stark unit...
galois of reducible polynomialrootsof1 (bug1)(%c,varn=%ld,prec=%ld,valp=%ld):) ^{%ld}not an integral modelelement_mulid%simplify_iinvalid data in qfbevaltwistpartialzeta (3), j = %ld/%ldnfissplit  1) T_2 bound for %s: %Z
inseparable relative equation in rnfequationROUND2: epsilon = %ld	avma = %ld
plot(X=a,b,expr,{ymin},{ymax}): crude plot of expression expr, X goes from a to b, with Y ranging from ymin to ymax. If ymin (resp. ymax) is not given, the minima (resp. the maxima) of the expression is used insteadplotdraw(list, {flag=0}): draw vector of rectwindows list at indicated x,y positions; list is a vector w1,x1,y1,w2,x2,y2,etc. . If flag!=0, x1, y1 etc. express fractions of the size of the current output deviceplotpointsize(w,size): change the "size" of following points in rectwindow w. w=-1 changes global valueplottermplotterm("termname"): set terminal to plot in high resolution to. Ignored by some drivers. In gnuplot driver possible terminals are the same as in gnuplot, terminal options can be put after the terminal name and space; terminal size can be put immediately after the name, as in "gif=300,200". If term is "?", lists possible values. Positive return value means successincorrect ideal in idealtypideals don't sum to Z_K in idealaddmultooneincompatible variables in idealinvtoo many iterations for desired precision in integration routinenegative exponent%s %%!
50 50 translate
/p {moveto 0 2 rlineto 2 0 rlineto 0 -2 rlineto closepath fill} def
/l {lineto} def
/m {moveto} def
/Times-Roman findfont %ld scalefont setfont
%g %g scale
ECM: number too small to justify this stage
	found factor = %Z
OddPwrs: is %Z
	...a power?
	modulo: resid. (remaining possibilities)
[2] ifac_decompnon-existent factor class in ifac_mainIFAC: factor %Z
	is prime
 (so far)...MPQS: got %ld factors%s
MPQS: decomposed a square

ns = %ld
vandermondeinverseA4GaloisConj:tau=%Z 
Trying degre %d.
GaloisConj:Testing %ZGaloisIndex: Using hash value w=%ld
mathnfspec with large entriesB = %Z
    hnflll done
X11Cannot reset output routines back...Gnuplotqfi_unit_by_disc (%ld)lllfp[1]isanypowerexponent overflow in regulaepsilon_3 -> %Z
  B0  = %Z
NaN or Infinity in dbltorJd+zO<++˛˛˛˛˛(3L[4DBBBתBתȪ5$+!hHXzzzzzzzzzzzzzzzzz,,*,,,,?<<]6M=====0==nQ/><>4K4444444444444444%$='X99===='''==

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